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        Module&nbsp;Numerics
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<h1 class="epydoc">Source Code for <a href="dadi.Numerics-module.html">Module dadi.Numerics</a></h1>
<pre class="py-src">
<a name="L1"></a><tt class="py-lineno">  1</tt>  <tt class="py-line"><tt class="py-docstring">"""</tt> </tt>
<a name="L2"></a><tt class="py-lineno">  2</tt>  <tt class="py-line"><tt class="py-docstring">Numerically useful functions, including extrapolation and default grid.</tt> </tt>
<a name="L3"></a><tt class="py-lineno">  3</tt>  <tt class="py-line"><tt class="py-docstring">"""</tt> </tt>
<a name="L4"></a><tt class="py-lineno">  4</tt>  <tt class="py-line"><tt class="py-keyword">import</tt> <tt class="py-name">logging</tt> </tt>
<a name="L5"></a><tt class="py-lineno">  5</tt>  <tt class="py-line"><tt id="link-0" class="py-name" targets="Variable dadi.Hessian.logger=dadi.Hessian-module.html#logger,Variable dadi.Inference.logger=dadi.Inference-module.html#logger,Variable dadi.Integration.logger=dadi.Integration-module.html#logger,Variable dadi.Numerics.logger=dadi.Numerics-module.html#logger,Variable dadi.RunInParallel.logger=dadi.RunInParallel-module.html#logger,Variable dadi.Spectrum_mod.logger=dadi.Spectrum_mod-module.html#logger"><a title="dadi.Hessian.logger
dadi.Inference.logger
dadi.Integration.logger
dadi.Numerics.logger
dadi.RunInParallel.logger
dadi.Spectrum_mod.logger" class="py-name" href="#" onclick="return doclink('link-0', 'logger', 'link-0');">logger</a></tt> <tt class="py-op">=</tt> <tt class="py-name">logging</tt><tt class="py-op">.</tt><tt class="py-name">getLogger</tt><tt class="py-op">(</tt><tt class="py-string">'Numerics'</tt><tt class="py-op">)</tt> </tt>
<a name="L6"></a><tt class="py-lineno">  6</tt>  <tt class="py-line"> </tt>
<a name="L7"></a><tt class="py-lineno">  7</tt>  <tt class="py-line"><tt class="py-keyword">import</tt> <tt class="py-name">functools</tt><tt class="py-op">,</tt> <tt class="py-name">os</tt> </tt>
<a name="L8"></a><tt class="py-lineno">  8</tt>  <tt class="py-line"><tt class="py-keyword">import</tt> <tt class="py-name">numpy</tt> </tt>
<a name="L9"></a><tt class="py-lineno">  9</tt>  <tt class="py-line"><tt class="py-comment"># Account for difference in scipy installations.</tt> </tt>
<a name="L10"></a><tt class="py-lineno"> 10</tt>  <tt class="py-line"><tt class="py-keyword">try</tt><tt class="py-op">:</tt> </tt>
<a name="L11"></a><tt class="py-lineno"> 11</tt>  <tt class="py-line">    <tt class="py-keyword">from</tt> <tt class="py-name">scipy</tt><tt class="py-op">.</tt><tt class="py-name">misc</tt> <tt class="py-keyword">import</tt> <tt class="py-name">comb</tt> </tt>
<a name="L12"></a><tt class="py-lineno"> 12</tt>  <tt class="py-line"><tt class="py-keyword">except</tt> <tt class="py-name">ImportError</tt><tt class="py-op">:</tt> </tt>
<a name="L13"></a><tt class="py-lineno"> 13</tt>  <tt class="py-line">    <tt class="py-keyword">from</tt> <tt class="py-name">scipy</tt> <tt class="py-keyword">import</tt> <tt class="py-name">comb</tt> </tt>
<a name="L14"></a><tt class="py-lineno"> 14</tt>  <tt class="py-line"><tt class="py-keyword">from</tt> <tt class="py-name">scipy</tt><tt class="py-op">.</tt><tt class="py-name">special</tt> <tt class="py-keyword">import</tt> <tt class="py-name">gammaln</tt> </tt>
<a name="L15"></a><tt class="py-lineno"> 15</tt>  <tt class="py-line"> </tt>
<a name="quadratic_grid"></a><div id="quadratic_grid-def"><a name="L16"></a><tt class="py-lineno"> 16</tt> <a class="py-toggle" href="#" id="quadratic_grid-toggle" onclick="return toggle('quadratic_grid');">-</a><tt class="py-line"><tt class="py-keyword">def</tt> <a class="py-def-name" href="dadi.Numerics-module.html#quadratic_grid">quadratic_grid</a><tt class="py-op">(</tt><tt class="py-param">num_pts</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
</div><div id="quadratic_grid-collapsed" style="display:none;" pad="+++" indent="++++"></div><div id="quadratic_grid-expanded"><a name="L17"></a><tt class="py-lineno"> 17</tt>  <tt class="py-line">    <tt class="py-docstring">"""</tt> </tt>
<a name="L18"></a><tt class="py-lineno"> 18</tt>  <tt class="py-line"><tt class="py-docstring">    A nonuniform grid of points on [0,1] with a quadratic pattern of spacings.</tt> </tt>
<a name="L19"></a><tt class="py-lineno"> 19</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L20"></a><tt class="py-lineno"> 20</tt>  <tt class="py-line"><tt class="py-docstring">    The grid is weighted to be denser near 0 and 1, which is useful for</tt> </tt>
<a name="L21"></a><tt class="py-lineno"> 21</tt>  <tt class="py-line"><tt class="py-docstring">    population genetic simulations. In between, it smoothly increases and</tt> </tt>
<a name="L22"></a><tt class="py-lineno"> 22</tt>  <tt class="py-line"><tt class="py-docstring">    then decreases the step size.</tt> </tt>
<a name="L23"></a><tt class="py-lineno"> 23</tt>  <tt class="py-line"><tt class="py-docstring">    """</tt> </tt>
<a name="L24"></a><tt class="py-lineno"> 24</tt>  <tt class="py-line">    <tt class="py-comment"># Rounds down...</tt> </tt>
<a name="L25"></a><tt class="py-lineno"> 25</tt>  <tt class="py-line">    <tt class="py-name">small_pts</tt> <tt class="py-op">=</tt> <tt class="py-name">int</tt><tt class="py-op">(</tt><tt class="py-name">num_pts</tt> <tt class="py-op">/</tt> <tt class="py-number">10</tt><tt class="py-op">)</tt> </tt>
<a name="L26"></a><tt class="py-lineno"> 26</tt>  <tt class="py-line">    <tt class="py-name">large_pts</tt> <tt class="py-op">=</tt> <tt class="py-name">num_pts</tt> <tt class="py-op">-</tt> <tt class="py-name">small_pts</tt><tt class="py-op">+</tt><tt class="py-number">1</tt> </tt>
<a name="L27"></a><tt class="py-lineno"> 27</tt>  <tt class="py-line"> </tt>
<a name="L28"></a><tt class="py-lineno"> 28</tt>  <tt class="py-line">    <tt class="py-name">grid</tt> <tt class="py-op">=</tt> <tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">linspace</tt><tt class="py-op">(</tt><tt class="py-number">0</tt><tt class="py-op">,</tt> <tt class="py-number">0.05</tt><tt class="py-op">,</tt> <tt class="py-name">small_pts</tt><tt class="py-op">)</tt> </tt>
<a name="L29"></a><tt class="py-lineno"> 29</tt>  <tt class="py-line"> </tt>
<a name="L30"></a><tt class="py-lineno"> 30</tt>  <tt class="py-line">    <tt class="py-comment"># The interior grid spacings are a quadratic function of x, being</tt> </tt>
<a name="L31"></a><tt class="py-lineno"> 31</tt>  <tt class="py-line">    <tt class="py-comment"># approximately x1 at the boundaries. To achieve this, our actual grid</tt> </tt>
<a name="L32"></a><tt class="py-lineno"> 32</tt>  <tt class="py-line">    <tt class="py-comment"># positions must come from a cubic function.</tt> </tt>
<a name="L33"></a><tt class="py-lineno"> 33</tt>  <tt class="py-line">    <tt class="py-comment"># Here I calculate and apply the appropriate conditions:</tt> </tt>
<a name="L34"></a><tt class="py-lineno"> 34</tt>  <tt class="py-line">    <tt class="py-comment">#   x[q = 0] = x_start  =&gt;  d = x_start</tt> </tt>
<a name="L35"></a><tt class="py-lineno"> 35</tt>  <tt class="py-line">    <tt class="py-comment">#   dx/dq [q = 0] = dx_start =&gt; c = dx_start/dq</tt> </tt>
<a name="L36"></a><tt class="py-lineno"> 36</tt>  <tt class="py-line">    <tt class="py-comment">#   x[q = 1] = 1</tt> </tt>
<a name="L37"></a><tt class="py-lineno"> 37</tt>  <tt class="py-line">    <tt class="py-comment">#   dx/dq [q = 1] = dx_start</tt> </tt>
<a name="L38"></a><tt class="py-lineno"> 38</tt>  <tt class="py-line">     </tt>
<a name="L39"></a><tt class="py-lineno"> 39</tt>  <tt class="py-line">    <tt class="py-name">q</tt> <tt class="py-op">=</tt> <tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">linspace</tt><tt class="py-op">(</tt><tt class="py-number">0</tt><tt class="py-op">,</tt> <tt class="py-number">1</tt><tt class="py-op">,</tt> <tt class="py-name">large_pts</tt><tt class="py-op">)</tt> </tt>
<a name="L40"></a><tt class="py-lineno"> 40</tt>  <tt class="py-line">    <tt class="py-name">dq</tt> <tt class="py-op">=</tt> <tt class="py-name">q</tt><tt class="py-op">[</tt><tt class="py-number">1</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">q</tt><tt class="py-op">[</tt><tt class="py-number">0</tt><tt class="py-op">]</tt> </tt>
<a name="L41"></a><tt class="py-lineno"> 41</tt>  <tt class="py-line">    <tt class="py-name">x_start</tt> <tt class="py-op">=</tt> <tt class="py-name">grid</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt> </tt>
<a name="L42"></a><tt class="py-lineno"> 42</tt>  <tt class="py-line">    <tt class="py-name">dx_start</tt> <tt class="py-op">=</tt> <tt class="py-name">grid</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">grid</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">2</tt><tt class="py-op">]</tt> </tt>
<a name="L43"></a><tt class="py-lineno"> 43</tt>  <tt class="py-line"> </tt>
<a name="L44"></a><tt class="py-lineno"> 44</tt>  <tt class="py-line">    <tt class="py-name">d</tt> <tt class="py-op">=</tt> <tt class="py-name">x_start</tt> </tt>
<a name="L45"></a><tt class="py-lineno"> 45</tt>  <tt class="py-line">    <tt class="py-name">c</tt> <tt class="py-op">=</tt> <tt class="py-name">dx_start</tt><tt class="py-op">/</tt><tt class="py-name">dq</tt> </tt>
<a name="L46"></a><tt class="py-lineno"> 46</tt>  <tt class="py-line">    <tt class="py-name">b</tt> <tt class="py-op">=</tt> <tt class="py-op">-</tt><tt class="py-number">3</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-name">dq</tt> <tt class="py-op">+</tt> <tt class="py-name">dx_start</tt> <tt class="py-op">+</tt> <tt class="py-name">dq</tt><tt class="py-op">*</tt><tt class="py-name">x_start</tt><tt class="py-op">)</tt><tt class="py-op">/</tt><tt class="py-name">dq</tt> </tt>
<a name="L47"></a><tt class="py-lineno"> 47</tt>  <tt class="py-line">    <tt class="py-name">a</tt> <tt class="py-op">=</tt> <tt class="py-op">-</tt><tt class="py-number">2</tt> <tt class="py-op">*</tt> <tt class="py-name">b</tt><tt class="py-op">/</tt><tt class="py-number">3</tt> </tt>
<a name="L48"></a><tt class="py-lineno"> 48</tt>  <tt class="py-line">    <tt class="py-name">grid</tt> <tt class="py-op">=</tt> <tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">concatenate</tt><tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-name">grid</tt><tt class="py-op">[</tt><tt class="py-op">:</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">,</tt> <tt class="py-name">a</tt><tt class="py-op">*</tt><tt class="py-name">q</tt><tt class="py-op">**</tt><tt class="py-number">3</tt> <tt class="py-op">+</tt> <tt class="py-name">b</tt><tt class="py-op">*</tt><tt class="py-name">q</tt><tt class="py-op">**</tt><tt class="py-number">2</tt> <tt class="py-op">+</tt> <tt class="py-name">c</tt><tt class="py-op">*</tt><tt class="py-name">q</tt> <tt class="py-op">+</tt> <tt class="py-name">d</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> </tt>
<a name="L49"></a><tt class="py-lineno"> 49</tt>  <tt class="py-line"> </tt>
<a name="L50"></a><tt class="py-lineno"> 50</tt>  <tt class="py-line">    <tt class="py-keyword">return</tt> <tt class="py-name">grid</tt> </tt>
</div><a name="L51"></a><tt class="py-lineno"> 51</tt>  <tt class="py-line"> </tt>
<a name="estimate_best_exp_grid_crwd"></a><div id="estimate_best_exp_grid_crwd-def"><a name="L52"></a><tt class="py-lineno"> 52</tt> <a class="py-toggle" href="#" id="estimate_best_exp_grid_crwd-toggle" onclick="return toggle('estimate_best_exp_grid_crwd');">-</a><tt class="py-line"><tt class="py-keyword">def</tt> <a class="py-def-name" href="dadi.Numerics-module.html#estimate_best_exp_grid_crwd">estimate_best_exp_grid_crwd</a><tt class="py-op">(</tt><tt class="py-param">ns</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
</div><div id="estimate_best_exp_grid_crwd-collapsed" style="display:none;" pad="+++" indent="++++"></div><div id="estimate_best_exp_grid_crwd-expanded"><a name="L53"></a><tt class="py-lineno"> 53</tt>  <tt class="py-line">    <tt class="py-docstring">"""</tt> </tt>
<a name="L54"></a><tt class="py-lineno"> 54</tt>  <tt class="py-line"><tt class="py-docstring">    Emperical "best" values for exponential grid crowding.</tt> </tt>
<a name="L55"></a><tt class="py-lineno"> 55</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L56"></a><tt class="py-lineno"> 56</tt>  <tt class="py-line"><tt class="py-docstring">    These functional forms were estimated by running many simulations at</tt> </tt>
<a name="L57"></a><tt class="py-lineno"> 57</tt>  <tt class="py-line"><tt class="py-docstring">    different parameter values and asking when a simulation at pts_l = [max(ns),</tt> </tt>
<a name="L58"></a><tt class="py-lineno"> 58</tt>  <tt class="py-line"><tt class="py-docstring">    max(ns)+10, max(ns)+20] was most accurate.</tt> </tt>
<a name="L59"></a><tt class="py-lineno"> 59</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L60"></a><tt class="py-lineno"> 60</tt>  <tt class="py-line"><tt class="py-docstring">    These cannot be considered absolute best values, as that may depend on the</tt> </tt>
<a name="L61"></a><tt class="py-lineno"> 61</tt>  <tt class="py-line"><tt class="py-docstring">    model. It does seem broadly true that the optimal value of crwd increases</tt> </tt>
<a name="L62"></a><tt class="py-lineno"> 62</tt>  <tt class="py-line"><tt class="py-docstring">    with system size, up to a point.</tt> </tt>
<a name="L63"></a><tt class="py-lineno"> 63</tt>  <tt class="py-line"><tt class="py-docstring">    """</tt> </tt>
<a name="L64"></a><tt class="py-lineno"> 64</tt>  <tt class="py-line">    <tt class="py-name">xx</tt> <tt class="py-op">=</tt> <tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">mean</tt><tt class="py-op">(</tt><tt class="py-name">ns</tt><tt class="py-op">)</tt> </tt>
<a name="L65"></a><tt class="py-lineno"> 65</tt>  <tt class="py-line">    <tt class="py-keyword">if</tt> <tt class="py-name">len</tt><tt class="py-op">(</tt><tt class="py-name">ns</tt><tt class="py-op">)</tt> <tt class="py-op">==</tt> <tt class="py-number">1</tt><tt class="py-op">:</tt> </tt>
<a name="L66"></a><tt class="py-lineno"> 66</tt>  <tt class="py-line">        <tt class="py-keyword">return</tt> <tt class="py-name">min</tt><tt class="py-op">(</tt><tt class="py-name">max</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">**</tt><tt class="py-number">0.4</tt> <tt class="py-op">/</tt> <tt class="py-number">1.3</tt><tt class="py-op">,</tt> <tt class="py-number">1</tt><tt class="py-op">)</tt><tt class="py-op">,</tt> <tt class="py-number">9</tt><tt class="py-op">)</tt> </tt>
<a name="L67"></a><tt class="py-lineno"> 67</tt>  <tt class="py-line">    <tt class="py-keyword">elif</tt> <tt class="py-name">len</tt><tt class="py-op">(</tt><tt class="py-name">ns</tt><tt class="py-op">)</tt> <tt class="py-op">==</tt> <tt class="py-number">2</tt><tt class="py-op">:</tt> </tt>
<a name="L68"></a><tt class="py-lineno"> 68</tt>  <tt class="py-line">        <tt class="py-keyword">return</tt> <tt class="py-name">min</tt><tt class="py-op">(</tt><tt class="py-name">max</tt><tt class="py-op">(</tt><tt class="py-number">1.5</tt> <tt class="py-op">*</tt> <tt class="py-name">xx</tt><tt class="py-op">**</tt><tt class="py-number">0.4</tt><tt class="py-op">,</tt> <tt class="py-number">1</tt><tt class="py-op">)</tt><tt class="py-op">,</tt> <tt class="py-number">8</tt><tt class="py-op">)</tt> </tt>
<a name="L69"></a><tt class="py-lineno"> 69</tt>  <tt class="py-line">    <tt class="py-keyword">else</tt><tt class="py-op">:</tt> </tt>
<a name="L70"></a><tt class="py-lineno"> 70</tt>  <tt class="py-line">        <tt class="py-keyword">raise</tt> <tt class="py-name">ValueError</tt><tt class="py-op">(</tt><tt class="py-string">'Due to computational expense, no optimum has been '</tt> </tt>
<a name="L71"></a><tt class="py-lineno"> 71</tt>  <tt class="py-line">                         <tt class="py-string">'determined for 3D or more models. Try sticking with '</tt> </tt>
<a name="L72"></a><tt class="py-lineno"> 72</tt>  <tt class="py-line">                         <tt class="py-string">'crwd=8.'</tt><tt class="py-op">)</tt> </tt>
</div><a name="L73"></a><tt class="py-lineno"> 73</tt>  <tt class="py-line"> </tt>
<a name="exponential_grid"></a><div id="exponential_grid-def"><a name="L74"></a><tt class="py-lineno"> 74</tt> <a class="py-toggle" href="#" id="exponential_grid-toggle" onclick="return toggle('exponential_grid');">-</a><tt class="py-line"><tt class="py-keyword">def</tt> <a class="py-def-name" href="dadi.Numerics-module.html#exponential_grid">exponential_grid</a><tt class="py-op">(</tt><tt class="py-param">pts</tt><tt class="py-op">,</tt> <tt class="py-param">crwd</tt><tt class="py-op">=</tt><tt class="py-number">8.</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
</div><div id="exponential_grid-collapsed" style="display:none;" pad="+++" indent="++++"></div><div id="exponential_grid-expanded"><a name="L75"></a><tt class="py-lineno"> 75</tt>  <tt class="py-line">    <tt class="py-docstring">"""</tt> </tt>
<a name="L76"></a><tt class="py-lineno"> 76</tt>  <tt class="py-line"><tt class="py-docstring">    An exponentially spaced grid. This is now the default grid.</tt> </tt>
<a name="L77"></a><tt class="py-lineno"> 77</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L78"></a><tt class="py-lineno"> 78</tt>  <tt class="py-line"><tt class="py-docstring">    crwd controls the degree to which grid points crowd against x=0 or x=1.</tt> </tt>
<a name="L79"></a><tt class="py-lineno"> 79</tt>  <tt class="py-line"><tt class="py-docstring">    The value of crwd=8 seems to be a good default for integration with large</tt> </tt>
<a name="L80"></a><tt class="py-lineno"> 80</tt>  <tt class="py-line"><tt class="py-docstring">    systems. See estimate_best_exp_grid_crwd for some empirical optimizations</tt> </tt>
<a name="L81"></a><tt class="py-lineno"> 81</tt>  <tt class="py-line"><tt class="py-docstring">    beyond this.</tt> </tt>
<a name="L82"></a><tt class="py-lineno"> 82</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L83"></a><tt class="py-lineno"> 83</tt>  <tt class="py-line"><tt class="py-docstring">    This grid was contributed by Simon Gravel.</tt> </tt>
<a name="L84"></a><tt class="py-lineno"> 84</tt>  <tt class="py-line"><tt class="py-docstring">    """</tt> </tt>
<a name="L85"></a><tt class="py-lineno"> 85</tt>  <tt class="py-line">    <tt class="py-name">unif</tt> <tt class="py-op">=</tt> <tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">linspace</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">,</tt><tt class="py-number">1</tt><tt class="py-op">,</tt><tt class="py-name">pts</tt><tt class="py-op">)</tt> </tt>
<a name="L86"></a><tt class="py-lineno"> 86</tt>  <tt class="py-line">    <tt class="py-name">grid</tt> <tt class="py-op">=</tt> <tt class="py-number">1.</tt><tt class="py-op">/</tt><tt class="py-op">(</tt><tt class="py-number">1.</tt> <tt class="py-op">+</tt> <tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">exp</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-name">crwd</tt><tt class="py-op">*</tt><tt class="py-name">unif</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> </tt>
<a name="L87"></a><tt class="py-lineno"> 87</tt>  <tt class="py-line"> </tt>
<a name="L88"></a><tt class="py-lineno"> 88</tt>  <tt class="py-line">    <tt class="py-comment"># Normalize</tt> </tt>
<a name="L89"></a><tt class="py-lineno"> 89</tt>  <tt class="py-line">    <tt class="py-name">grid</tt> <tt class="py-op">=</tt> <tt class="py-op">(</tt><tt class="py-name">grid</tt><tt class="py-op">-</tt><tt class="py-name">grid</tt><tt class="py-op">[</tt><tt class="py-number">0</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">/</tt><tt class="py-op">(</tt><tt class="py-name">grid</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">-</tt><tt class="py-name">grid</tt><tt class="py-op">[</tt><tt class="py-number">0</tt><tt class="py-op">]</tt><tt class="py-op">)</tt> </tt>
<a name="L90"></a><tt class="py-lineno"> 90</tt>  <tt class="py-line">    <tt class="py-keyword">return</tt> <tt class="py-name">grid</tt> </tt>
</div><a name="L91"></a><tt class="py-lineno"> 91</tt>  <tt class="py-line"> </tt>
<a name="L92"></a><tt class="py-lineno"> 92</tt>  <tt class="py-line"><tt class="py-name">default_grid</tt> <tt class="py-op">=</tt> <tt id="link-1" class="py-name" targets="Function dadi.Numerics.exponential_grid()=dadi.Numerics-module.html#exponential_grid"><a title="dadi.Numerics.exponential_grid" class="py-name" href="#" onclick="return doclink('link-1', 'exponential_grid', 'link-1');">exponential_grid</a></tt> </tt>
<a name="L93"></a><tt class="py-lineno"> 93</tt>  <tt class="py-line"> </tt>
<a name="end_point_first_derivs"></a><div id="end_point_first_derivs-def"><a name="L94"></a><tt class="py-lineno"> 94</tt> <a class="py-toggle" href="#" id="end_point_first_derivs-toggle" onclick="return toggle('end_point_first_derivs');">-</a><tt class="py-line"><tt class="py-keyword">def</tt> <a class="py-def-name" href="dadi.Numerics-module.html#end_point_first_derivs">end_point_first_derivs</a><tt class="py-op">(</tt><tt class="py-param">xx</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
</div><div id="end_point_first_derivs-collapsed" style="display:none;" pad="+++" indent="++++"></div><div id="end_point_first_derivs-expanded"><a name="L95"></a><tt class="py-lineno"> 95</tt>  <tt class="py-line">    <tt class="py-docstring">"""</tt> </tt>
<a name="L96"></a><tt class="py-lineno"> 96</tt>  <tt class="py-line"><tt class="py-docstring">    Coefficients for a 5-point one-sided approximation of the first derivative.</tt> </tt>
<a name="L97"></a><tt class="py-lineno"> 97</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L98"></a><tt class="py-lineno"> 98</tt>  <tt class="py-line"><tt class="py-docstring">    xx: grid on which the data to be differentiated lives</tt> </tt>
<a name="L99"></a><tt class="py-lineno"> 99</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L100"></a><tt class="py-lineno">100</tt>  <tt class="py-line"><tt class="py-docstring">    Returns ret, a 2x5 array. ret[0] is the coefficients for an approximation</tt> </tt>
<a name="L101"></a><tt class="py-lineno">101</tt>  <tt class="py-line"><tt class="py-docstring">    of the derivative at xx[0]. It is used by deriv = numpy.dot(ret[0],</tt> </tt>
<a name="L102"></a><tt class="py-lineno">102</tt>  <tt class="py-line"><tt class="py-docstring">    yy[:5]). ret[1] is the coefficients for the derivative at xx[-1]. It can be</tt> </tt>
<a name="L103"></a><tt class="py-lineno">103</tt>  <tt class="py-line"><tt class="py-docstring">    used by deriv = dot(ret[1][::-1], yy[-5:]). (Note that we need to reverse</tt> </tt>
<a name="L104"></a><tt class="py-lineno">104</tt>  <tt class="py-line"><tt class="py-docstring">    the coefficient array here.</tt> </tt>
<a name="L105"></a><tt class="py-lineno">105</tt>  <tt class="py-line"><tt class="py-docstring">    """</tt> </tt>
<a name="L106"></a><tt class="py-lineno">106</tt>  <tt class="py-line">    <tt class="py-name">output</tt> <tt class="py-op">=</tt> <tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">zeros</tt><tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-number">2</tt><tt class="py-op">,</tt><tt class="py-number">5</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> </tt>
<a name="L107"></a><tt class="py-lineno">107</tt>  <tt class="py-line"> </tt>
<a name="L108"></a><tt class="py-lineno">108</tt>  <tt class="py-line">    <tt class="py-comment"># These are the coefficients for a one-sided 1st derivative of f[0].</tt> </tt>
<a name="L109"></a><tt class="py-lineno">109</tt>  <tt class="py-line">    <tt class="py-comment"># So f'[0] = d10_0 f[0] + d10_1 f[1] + d10_2 f[2] + d10_3 f[3] + d10_4 f[4]</tt> </tt>
<a name="L110"></a><tt class="py-lineno">110</tt>  <tt class="py-line">    <tt class="py-comment"># This expression is 4th order accurate.</tt> </tt>
<a name="L111"></a><tt class="py-lineno">111</tt>  <tt class="py-line">    <tt class="py-comment"># To derive it in Mathematica, use NDSolve`FiniteDifferenceDerivative[1, {xx[0], xx[1], xx[2], xx[3], xx[4]}, {f[0], f[1], f[2], f[3], f[4]}, DifferenceOrder -&gt; 4][[1]] // Simplify</tt> </tt>
<a name="L112"></a><tt class="py-lineno">112</tt>  <tt class="py-line">    <tt class="py-name">d10_0</tt> <tt class="py-op">=</tt> <tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-number">1</tt> <tt class="py-op">+</tt> <tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-number">1</tt> <tt class="py-op">+</tt> <tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-number">2</tt><tt class="py-op">*</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">0</tt><tt class="py-op">]</tt> <tt class="py-op">+</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">1</tt><tt class="py-op">]</tt> <tt class="py-op">+</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">2</tt><tt class="py-op">]</tt><tt class="py-op">)</tt> <tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">0</tt><tt class="py-op">]</tt> <tt class="py-op">+</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">3</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">/</tt> <tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">0</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">0</tt><tt class="py-op">]</tt> <tt class="py-op">+</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">2</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">0</tt><tt class="py-op">]</tt> <tt class="py-op">+</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">4</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">/</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">0</tt><tt class="py-op">]</tt> <tt class="py-op">+</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">3</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">/</tt> <tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">0</tt><tt class="py-op">]</tt> <tt class="py-op">+</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">4</tt><tt class="py-op">]</tt><tt class="py-op">)</tt> </tt>
<a name="L113"></a><tt class="py-lineno">113</tt>  <tt class="py-line">    <tt class="py-name">d10_1</tt> <tt class="py-op">=</tt> <tt class="py-op">-</tt><tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">0</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">2</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">0</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">3</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">0</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">4</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">/</tt> <tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">0</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">1</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">2</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">1</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">3</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">1</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">4</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> </tt>
<a name="L114"></a><tt class="py-lineno">114</tt>  <tt class="py-line">    <tt class="py-name">d10_2</tt> <tt class="py-op">=</tt> <tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">0</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">0</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">3</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">0</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">4</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">/</tt> <tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">0</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">2</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">1</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">2</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">2</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">3</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">2</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">4</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> </tt>
<a name="L115"></a><tt class="py-lineno">115</tt>  <tt class="py-line">    <tt class="py-name">d10_3</tt> <tt class="py-op">=</tt> <tt class="py-op">-</tt><tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">0</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">0</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">2</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">0</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">4</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">/</tt> <tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">0</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">3</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">1</tt><tt class="py-op">]</tt> <tt class="py-op">+</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">3</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">2</tt><tt class="py-op">]</tt> <tt class="py-op">+</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">3</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">3</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">4</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> </tt>
<a name="L116"></a><tt class="py-lineno">116</tt>  <tt class="py-line">    <tt class="py-name">d10_4</tt> <tt class="py-op">=</tt> <tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">0</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">0</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">2</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">0</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">3</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">/</tt> <tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">0</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">4</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">1</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">4</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">2</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">4</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">3</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-number">4</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> </tt>
<a name="L117"></a><tt class="py-lineno">117</tt>  <tt class="py-line"> </tt>
<a name="L118"></a><tt class="py-lineno">118</tt>  <tt class="py-line">    <tt class="py-name">output</tt><tt class="py-op">[</tt><tt class="py-number">0</tt><tt class="py-op">]</tt> <tt class="py-op">=</tt> <tt class="py-op">(</tt><tt class="py-name">d10_0</tt><tt class="py-op">,</tt> <tt class="py-name">d10_1</tt><tt class="py-op">,</tt> <tt class="py-name">d10_2</tt><tt class="py-op">,</tt> <tt class="py-name">d10_3</tt><tt class="py-op">,</tt> <tt class="py-name">d10_4</tt><tt class="py-op">)</tt> </tt>
<a name="L119"></a><tt class="py-lineno">119</tt>  <tt class="py-line"> </tt>
<a name="L120"></a><tt class="py-lineno">120</tt>  <tt class="py-line">    <tt class="py-comment"># These are the coefficients for a one-sided 1st derivative of f[-1].</tt> </tt>
<a name="L121"></a><tt class="py-lineno">121</tt>  <tt class="py-line">    <tt class="py-comment"># So f'[-1] = d1m1_m1 f[-1] + d1m1_m2 f[-2] + d1m1_m3 f[-3] + d1m1_m4 f[-4]</tt> </tt>
<a name="L122"></a><tt class="py-lineno">122</tt>  <tt class="py-line">    <tt class="py-comment">#             + d1m1_m5 f[-5]</tt> </tt>
<a name="L123"></a><tt class="py-lineno">123</tt>  <tt class="py-line">    <tt class="py-name">d1m1_m1</tt> <tt class="py-op">=</tt> <tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-number">3</tt><tt class="py-op">*</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">2</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-number">4</tt><tt class="py-op">*</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt> <tt class="py-op">+</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">3</tt><tt class="py-op">]</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-number">2</tt><tt class="py-op">*</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">2</tt><tt class="py-op">]</tt> <tt class="py-op">+</tt> <tt class="py-number">3</tt><tt class="py-op">*</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">4</tt><tt class="py-op">]</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">3</tt><tt class="py-op">]</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">2</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-number">2</tt><tt class="py-op">*</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-number">2</tt><tt class="py-op">*</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">2</tt><tt class="py-op">]</tt> <tt class="py-op">+</tt> <tt class="py-number">3</tt><tt class="py-op">*</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">5</tt><tt class="py-op">]</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">3</tt><tt class="py-op">]</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">2</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-number">2</tt><tt class="py-op">*</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">4</tt><tt class="py-op">]</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">3</tt><tt class="py-op">]</tt> <tt class="py-op">+</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">2</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-number">2</tt><tt class="py-op">*</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-number">2</tt><tt class="py-op">*</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">2</tt><tt class="py-op">]</tt> <tt class="py-op">+</tt> <tt class="py-number">3</tt><tt class="py-op">*</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">/</tt> <tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">4</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">5</tt><tt class="py-op">]</tt> <tt class="py-op">+</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">3</tt><tt class="py-op">]</tt> <tt class="py-op">+</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">2</tt><tt class="py-op">]</tt> <tt class="py-op">+</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> </tt>
<a name="L124"></a><tt class="py-lineno">124</tt>  <tt class="py-line">    <tt class="py-name">d1m1_m2</tt> <tt class="py-op">=</tt> <tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">5</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">4</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">3</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">/</tt> <tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">5</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">2</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">4</tt><tt class="py-op">]</tt> <tt class="py-op">+</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">2</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">3</tt><tt class="py-op">]</tt> <tt class="py-op">+</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">2</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">2</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> </tt>
<a name="L125"></a><tt class="py-lineno">125</tt>  <tt class="py-line">    <tt class="py-name">d1m1_m3</tt> <tt class="py-op">=</tt> <tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">5</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">4</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">2</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">/</tt> <tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">5</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">3</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">4</tt><tt class="py-op">]</tt> <tt class="py-op">+</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">3</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">3</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">2</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">3</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> </tt>
<a name="L126"></a><tt class="py-lineno">126</tt>  <tt class="py-line">    <tt class="py-name">d1m1_m4</tt> <tt class="py-op">=</tt> <tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">5</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">3</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">2</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">/</tt> <tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">5</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">4</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">4</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">3</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">4</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">2</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">4</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">)</tt>  </tt>
<a name="L127"></a><tt class="py-lineno">127</tt>  <tt class="py-line">    <tt class="py-name">d1m1_m5</tt> <tt class="py-op">=</tt> <tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">4</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">3</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">2</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">/</tt> <tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">5</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">4</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">5</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">3</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">5</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">2</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">5</tt><tt class="py-op">]</tt> <tt class="py-op">+</tt> <tt class="py-name">xx</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> </tt>
<a name="L128"></a><tt class="py-lineno">128</tt>  <tt class="py-line"> </tt>
<a name="L129"></a><tt class="py-lineno">129</tt>  <tt class="py-line">    <tt class="py-name">output</tt><tt class="py-op">[</tt><tt class="py-number">1</tt><tt class="py-op">]</tt> <tt class="py-op">=</tt> <tt class="py-op">(</tt><tt class="py-name">d1m1_m1</tt><tt class="py-op">,</tt> <tt class="py-name">d1m1_m2</tt><tt class="py-op">,</tt> <tt class="py-name">d1m1_m3</tt><tt class="py-op">,</tt> <tt class="py-name">d1m1_m4</tt><tt class="py-op">,</tt> <tt class="py-name">d1m1_m5</tt><tt class="py-op">)</tt> </tt>
<a name="L130"></a><tt class="py-lineno">130</tt>  <tt class="py-line"> </tt>
<a name="L131"></a><tt class="py-lineno">131</tt>  <tt class="py-line">    <tt class="py-keyword">return</tt> <tt class="py-name">output</tt> </tt>
</div><a name="L132"></a><tt class="py-lineno">132</tt>  <tt class="py-line"> </tt>
<a name="linear_extrap"></a><div id="linear_extrap-def"><a name="L133"></a><tt class="py-lineno">133</tt> <a class="py-toggle" href="#" id="linear_extrap-toggle" onclick="return toggle('linear_extrap');">-</a><tt class="py-line"><tt class="py-keyword">def</tt> <a class="py-def-name" href="dadi.Numerics-module.html#linear_extrap">linear_extrap</a><tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-param">y1</tt><tt class="py-op">,</tt> <tt class="py-param">y2</tt><tt class="py-op">)</tt><tt class="py-op">,</tt> <tt class="py-op">(</tt><tt class="py-param">x1</tt><tt class="py-op">,</tt> <tt class="py-param">x2</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
</div><div id="linear_extrap-collapsed" style="display:none;" pad="+++" indent="++++"></div><div id="linear_extrap-expanded"><a name="L134"></a><tt class="py-lineno">134</tt>  <tt class="py-line">    <tt class="py-docstring">"""</tt> </tt>
<a name="L135"></a><tt class="py-lineno">135</tt>  <tt class="py-line"><tt class="py-docstring">    Linearly extrapolate from two x,y pairs to x = 0.</tt> </tt>
<a name="L136"></a><tt class="py-lineno">136</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L137"></a><tt class="py-lineno">137</tt>  <tt class="py-line"><tt class="py-docstring">    y1,y2: y values from x,y pairs. Note that these can be arrays of values.</tt> </tt>
<a name="L138"></a><tt class="py-lineno">138</tt>  <tt class="py-line"><tt class="py-docstring">    x1,x2: x values from x,y pairs. These should be scalars.</tt> </tt>
<a name="L139"></a><tt class="py-lineno">139</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L140"></a><tt class="py-lineno">140</tt>  <tt class="py-line"><tt class="py-docstring">    Returns extrapolated y at x=0.</tt> </tt>
<a name="L141"></a><tt class="py-lineno">141</tt>  <tt class="py-line"><tt class="py-docstring">    """</tt> </tt>
<a name="L142"></a><tt class="py-lineno">142</tt>  <tt class="py-line">    <tt class="py-keyword">return</tt> <tt class="py-op">(</tt><tt class="py-name">x2</tt> <tt class="py-op">*</tt> <tt class="py-name">y1</tt> <tt class="py-op">-</tt> <tt class="py-name">x1</tt> <tt class="py-op">*</tt> <tt class="py-name">y2</tt><tt class="py-op">)</tt><tt class="py-op">/</tt><tt class="py-op">(</tt><tt class="py-name">x2</tt> <tt class="py-op">-</tt> <tt class="py-name">x1</tt><tt class="py-op">)</tt> </tt>
</div><a name="L143"></a><tt class="py-lineno">143</tt>  <tt class="py-line"> </tt>
<a name="quadratic_extrap"></a><div id="quadratic_extrap-def"><a name="L144"></a><tt class="py-lineno">144</tt> <a class="py-toggle" href="#" id="quadratic_extrap-toggle" onclick="return toggle('quadratic_extrap');">-</a><tt class="py-line"><tt class="py-keyword">def</tt> <a class="py-def-name" href="dadi.Numerics-module.html#quadratic_extrap">quadratic_extrap</a><tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-param">y1</tt><tt class="py-op">,</tt> <tt class="py-param">y2</tt><tt class="py-op">,</tt> <tt class="py-param">y3</tt><tt class="py-op">)</tt><tt class="py-op">,</tt> <tt class="py-op">(</tt><tt class="py-param">x1</tt><tt class="py-op">,</tt> <tt class="py-param">x2</tt><tt class="py-op">,</tt> <tt class="py-param">x3</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
</div><div id="quadratic_extrap-collapsed" style="display:none;" pad="+++" indent="++++"></div><div id="quadratic_extrap-expanded"><a name="L145"></a><tt class="py-lineno">145</tt>  <tt class="py-line">    <tt class="py-docstring">"""</tt> </tt>
<a name="L146"></a><tt class="py-lineno">146</tt>  <tt class="py-line"><tt class="py-docstring">    Quadratically extrapolate from three x,y pairs to x = 0.</tt> </tt>
<a name="L147"></a><tt class="py-lineno">147</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L148"></a><tt class="py-lineno">148</tt>  <tt class="py-line"><tt class="py-docstring">    y1,y2,y3: y values from x,y pairs. Note that these can be arrays of values.</tt> </tt>
<a name="L149"></a><tt class="py-lineno">149</tt>  <tt class="py-line"><tt class="py-docstring">    x1,x2,x3: x values from x,y pairs. These should be scalars.</tt> </tt>
<a name="L150"></a><tt class="py-lineno">150</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L151"></a><tt class="py-lineno">151</tt>  <tt class="py-line"><tt class="py-docstring">    Returns extrapolated y at x=0.</tt> </tt>
<a name="L152"></a><tt class="py-lineno">152</tt>  <tt class="py-line"><tt class="py-docstring">    """</tt> </tt>
<a name="L153"></a><tt class="py-lineno">153</tt>  <tt class="py-line">    <tt class="py-keyword">return</tt> <tt class="py-name">x2</tt><tt class="py-op">*</tt><tt class="py-name">x3</tt><tt class="py-op">/</tt><tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">-</tt><tt class="py-name">x2</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">-</tt><tt class="py-name">x3</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> <tt class="py-op">*</tt> <tt class="py-name">y1</tt> <tt class="py-op">+</tt> <tt class="py-name">x1</tt><tt class="py-op">*</tt><tt class="py-name">x3</tt><tt class="py-op">/</tt><tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-name">x2</tt><tt class="py-op">-</tt><tt class="py-name">x1</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x2</tt><tt class="py-op">-</tt><tt class="py-name">x3</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> <tt class="py-op">*</tt> <tt class="py-name">y2</tt>\ </tt>
<a name="L154"></a><tt class="py-lineno">154</tt>  <tt class="py-line">            <tt class="py-op">+</tt> <tt class="py-name">x1</tt><tt class="py-op">*</tt><tt class="py-name">x2</tt><tt class="py-op">/</tt><tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-name">x3</tt><tt class="py-op">-</tt><tt class="py-name">x1</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x3</tt><tt class="py-op">-</tt><tt class="py-name">x2</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> <tt class="py-op">*</tt> <tt class="py-name">y3</tt> </tt>
</div><a name="L155"></a><tt class="py-lineno">155</tt>  <tt class="py-line"> </tt>
<a name="cubic_extrap"></a><div id="cubic_extrap-def"><a name="L156"></a><tt class="py-lineno">156</tt> <a class="py-toggle" href="#" id="cubic_extrap-toggle" onclick="return toggle('cubic_extrap');">-</a><tt class="py-line"><tt class="py-keyword">def</tt> <a class="py-def-name" href="dadi.Numerics-module.html#cubic_extrap">cubic_extrap</a><tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-param">y1</tt><tt class="py-op">,</tt> <tt class="py-param">y2</tt><tt class="py-op">,</tt> <tt class="py-param">y3</tt><tt class="py-op">,</tt> <tt class="py-param">y4</tt><tt class="py-op">)</tt><tt class="py-op">,</tt> <tt class="py-op">(</tt><tt class="py-param">x1</tt><tt class="py-op">,</tt> <tt class="py-param">x2</tt><tt class="py-op">,</tt> <tt class="py-param">x3</tt><tt class="py-op">,</tt> <tt class="py-param">x4</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
</div><div id="cubic_extrap-collapsed" style="display:none;" pad="+++" indent="++++"></div><div id="cubic_extrap-expanded"><a name="L157"></a><tt class="py-lineno">157</tt>  <tt class="py-line">    <tt class="py-docstring">"""</tt> </tt>
<a name="L158"></a><tt class="py-lineno">158</tt>  <tt class="py-line"><tt class="py-docstring">    Cubically extrapolate from three x,y pairs to x = 0.</tt> </tt>
<a name="L159"></a><tt class="py-lineno">159</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L160"></a><tt class="py-lineno">160</tt>  <tt class="py-line"><tt class="py-docstring">    y1,y2,y3: y values from x,y pairs. Note that these can be arrays of values.</tt> </tt>
<a name="L161"></a><tt class="py-lineno">161</tt>  <tt class="py-line"><tt class="py-docstring">    x1,x2,x3: x values from x,y pairs. These should be scalars.</tt> </tt>
<a name="L162"></a><tt class="py-lineno">162</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L163"></a><tt class="py-lineno">163</tt>  <tt class="py-line"><tt class="py-docstring">    Returns extrapolated y at x=0.</tt> </tt>
<a name="L164"></a><tt class="py-lineno">164</tt>  <tt class="py-line"><tt class="py-docstring">    """</tt> </tt>
<a name="L165"></a><tt class="py-lineno">165</tt>  <tt class="py-line">    <tt class="py-comment"># This horrid implementation came from using CForm in Mathematica.</tt> </tt>
<a name="L166"></a><tt class="py-lineno">166</tt>  <tt class="py-line">    <tt class="py-name">Power</tt> <tt class="py-op">=</tt> <tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">power</tt> </tt>
<a name="L167"></a><tt class="py-lineno">167</tt>  <tt class="py-line">    <tt class="py-keyword">return</tt> <tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt> <tt class="py-op">-</tt> <tt class="py-name">x3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">x3</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">x1</tt> <tt class="py-op">-</tt> <tt class="py-name">x4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x3</tt> <tt class="py-op">-</tt> <tt class="py-name">x4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">x4</tt><tt class="py-op">*</tt> <tt class="py-name">y2</tt> <tt class="py-op">+</tt> <tt class="py-name">x2</tt><tt class="py-op">*</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x4</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-op">(</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x3</tt><tt class="py-op">,</tt><tt class="py-number">3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y1</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x3</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">x4</tt><tt class="py-op">*</tt><tt class="py-name">y1</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">x1</tt> <tt class="py-op">-</tt> <tt class="py-name">x4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y3</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">x2</tt><tt class="py-op">*</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x3</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-name">x1</tt> <tt class="py-op">+</tt> <tt class="py-name">x3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-name">y4</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x2</tt><tt class="py-op">,</tt><tt class="py-number">3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">x4</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-op">(</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x3</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-name">y1</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">x3</tt><tt class="py-op">*</tt><tt class="py-name">x4</tt><tt class="py-op">*</tt><tt class="py-name">y1</tt> <tt class="py-op">+</tt> <tt class="py-name">x1</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt> <tt class="py-op">-</tt> <tt class="py-name">x4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y3</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">x1</tt><tt class="py-op">*</tt><tt class="py-name">x3</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-name">x1</tt> <tt class="py-op">+</tt> <tt class="py-name">x3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-name">y4</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x2</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">*</tt><tt class="py-name">x4</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x4</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y3</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x3</tt><tt class="py-op">,</tt><tt class="py-number">3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">x4</tt><tt class="py-op">*</tt><tt class="py-name">y1</tt> <tt class="py-op">-</tt> <tt class="py-name">x1</tt><tt class="py-op">*</tt><tt class="py-name">y4</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">x3</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-op">(</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x4</tt><tt class="py-op">,</tt><tt class="py-number">3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-name">y1</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">,</tt><tt class="py-number">3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y4</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">/</tt> <tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt> <tt class="py-op">-</tt> <tt class="py-name">x2</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt> <tt class="py-op">-</tt> <tt class="py-name">x3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">x2</tt> <tt class="py-op">-</tt> <tt class="py-name">x3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt> <tt class="py-op">-</tt> <tt class="py-name">x4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">x2</tt> <tt class="py-op">-</tt> <tt class="py-name">x4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x3</tt> <tt class="py-op">-</tt> <tt class="py-name">x4</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> </tt>
</div><a name="L168"></a><tt class="py-lineno">168</tt>  <tt class="py-line"> </tt>
<a name="quartic_extrap"></a><div id="quartic_extrap-def"><a name="L169"></a><tt class="py-lineno">169</tt> <a class="py-toggle" href="#" id="quartic_extrap-toggle" onclick="return toggle('quartic_extrap');">-</a><tt class="py-line"><tt class="py-keyword">def</tt> <a class="py-def-name" href="dadi.Numerics-module.html#quartic_extrap">quartic_extrap</a><tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-param">y1</tt><tt class="py-op">,</tt><tt class="py-param">y2</tt><tt class="py-op">,</tt><tt class="py-param">y3</tt><tt class="py-op">,</tt><tt class="py-param">y4</tt><tt class="py-op">,</tt><tt class="py-param">y5</tt><tt class="py-op">)</tt><tt class="py-op">,</tt> <tt class="py-op">(</tt><tt class="py-param">x1</tt><tt class="py-op">,</tt><tt class="py-param">x2</tt><tt class="py-op">,</tt><tt class="py-param">x3</tt><tt class="py-op">,</tt><tt class="py-param">x4</tt><tt class="py-op">,</tt><tt class="py-param">x5</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
</div><div id="quartic_extrap-collapsed" style="display:none;" pad="+++" indent="++++"></div><div id="quartic_extrap-expanded"><a name="L170"></a><tt class="py-lineno">170</tt>  <tt class="py-line">    <tt class="py-docstring">"""</tt> </tt>
<a name="L171"></a><tt class="py-lineno">171</tt>  <tt class="py-line"><tt class="py-docstring">    Quartically extrapolate from three x,y pairs to x = 0.</tt> </tt>
<a name="L172"></a><tt class="py-lineno">172</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L173"></a><tt class="py-lineno">173</tt>  <tt class="py-line"><tt class="py-docstring">    y1,y2...: y values from x,y pairs. Note that these can be arrays of values.</tt> </tt>
<a name="L174"></a><tt class="py-lineno">174</tt>  <tt class="py-line"><tt class="py-docstring">    x1,x2...: x values from x,y pairs. These should be scalars.</tt> </tt>
<a name="L175"></a><tt class="py-lineno">175</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L176"></a><tt class="py-lineno">176</tt>  <tt class="py-line"><tt class="py-docstring">    Returns extrapolated y at x=0.</tt> </tt>
<a name="L177"></a><tt class="py-lineno">177</tt>  <tt class="py-line"><tt class="py-docstring">    """</tt> </tt>
<a name="L178"></a><tt class="py-lineno">178</tt>  <tt class="py-line">    <tt class="py-comment"># This horrid implementation came from using CForm in Mathematica.</tt> </tt>
<a name="L179"></a><tt class="py-lineno">179</tt>  <tt class="py-line">    <tt class="py-name">Power</tt> <tt class="py-op">=</tt> <tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">power</tt> </tt>
<a name="L180"></a><tt class="py-lineno">180</tt>  <tt class="py-line">    <tt class="py-keyword">return</tt> <tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt> <tt class="py-op">-</tt> <tt class="py-name">x3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">x3</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt> <tt class="py-op">-</tt> <tt class="py-name">x4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x3</tt> <tt class="py-op">-</tt> <tt class="py-name">x4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">x4</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt> <tt class="py-op">-</tt> <tt class="py-name">x5</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">x3</tt> <tt class="py-op">-</tt> <tt class="py-name">x5</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x4</tt> <tt class="py-op">-</tt> <tt class="py-name">x5</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">x5</tt><tt class="py-op">*</tt><tt class="py-name">y2</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x2</tt><tt class="py-op">,</tt><tt class="py-number">4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt> <tt class="py-op">-</tt> <tt class="py-name">x4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">x4</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt> <tt class="py-op">-</tt> <tt class="py-name">x5</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">x4</tt> <tt class="py-op">-</tt> <tt class="py-name">x5</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">x5</tt><tt class="py-op">*</tt><tt class="py-name">y3</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x3</tt><tt class="py-op">,</tt><tt class="py-number">3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">*</tt><tt class="py-name">x5</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-name">x1</tt> <tt class="py-op">+</tt> <tt class="py-name">x5</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y4</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x4</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x5</tt><tt class="py-op">*</tt><tt class="py-name">y1</tt> <tt class="py-op">-</tt> <tt class="py-name">x1</tt><tt class="py-op">*</tt><tt class="py-name">y5</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">x4</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-op">(</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x5</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y1</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y5</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x3</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">*</tt><tt class="py-name">x5</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt> <tt class="py-op">-</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x5</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y4</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x4</tt><tt class="py-op">,</tt><tt class="py-number">3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-op">(</tt><tt 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class="py-op">*</tt><tt class="py-name">y1</tt> <tt class="py-op">-</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">,</tt><tt class="py-number">3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y5</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x3</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">,</tt><tt class="py-number">3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt> <tt class="py-op">-</tt> <tt class="py-name">x5</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x5</tt><tt class="py-op">,</tt><tt class="py-number">3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y4</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x4</tt><tt class="py-op">,</tt><tt class="py-number">4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-op">(</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x5</tt><tt class="py-op">,</tt><tt class="py-number">3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y1</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">,</tt><tt class="py-number">3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y5</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x4</tt><tt class="py-op">,</tt><tt class="py-number">3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x5</tt><tt class="py-op">,</tt><tt class="py-number">4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y1</tt> <tt class="py-op">-</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">,</tt><tt class="py-number">4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y5</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x3</tt><tt class="py-op">,</tt><tt class="py-number">3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x5</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x5</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y4</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x4</tt><tt class="py-op">,</tt><tt class="py-number">4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x5</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y1</tt> <tt class="py-op">-</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y5</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x4</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-op">(</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x5</tt><tt class="py-op">,</tt><tt class="py-number">4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y1</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">,</tt><tt class="py-number">4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y5</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x2</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">*</tt><tt class="py-name">x4</tt><tt class="py-op">*</tt><tt class="py-name">x5</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x4</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x5</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-name">x4</tt> <tt class="py-op">+</tt> <tt class="py-name">x5</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">,</tt><tt class="py-number">3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x4</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x5</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x4</tt><tt class="py-op">,</tt><tt class="py-number">3</tt><tt class="py-op">)</tt> <tt class="py-op">-</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x5</tt><tt class="py-op">,</tt><tt class="py-number">3</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y3</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x3</tt><tt class="py-op">,</tt><tt class="py-number">4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">*</tt><tt class="py-name">x5</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x5</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y4</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x4</tt><tt class="py-op">,</tt><tt class="py-number">3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x5</tt><tt class="py-op">*</tt><tt class="py-name">y1</tt> <tt class="py-op">-</tt> <tt class="py-name">x1</tt><tt class="py-op">*</tt><tt class="py-name">y5</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">x4</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-op">(</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x5</tt><tt class="py-op">,</tt><tt class="py-number">3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y1</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">,</tt><tt class="py-number">3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y5</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x3</tt><tt class="py-op">,</tt><tt class="py-number">3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">*</tt><tt class="py-name">x5</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">,</tt><tt class="py-number">3</tt><tt class="py-op">)</tt> <tt class="py-op">-</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x5</tt><tt class="py-op">,</tt><tt class="py-number">3</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y4</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x4</tt><tt class="py-op">,</tt><tt class="py-number">4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-op">(</tt><tt class="py-name">x5</tt><tt class="py-op">*</tt><tt class="py-name">y1</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">x1</tt><tt class="py-op">*</tt><tt class="py-name">y5</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">x4</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x5</tt><tt class="py-op">,</tt><tt class="py-number">4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y1</tt> <tt class="py-op">-</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">,</tt><tt class="py-number">4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y5</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">x3</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">,</tt><tt class="py-number">3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x5</tt><tt class="py-op">,</tt><tt class="py-number">3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-name">x1</tt> <tt class="py-op">+</tt> <tt class="py-name">x5</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y4</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x4</tt><tt class="py-op">,</tt><tt class="py-number">4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x5</tt><tt class="py-op">,</tt><tt class="py-number">3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y1</tt> <tt class="py-op">-</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">,</tt><tt class="py-number">3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y5</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x4</tt><tt class="py-op">,</tt><tt class="py-number">3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-op">(</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x5</tt><tt class="py-op">,</tt><tt class="py-number">4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y1</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">,</tt><tt class="py-number">4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y5</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">/</tt> <tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt> <tt class="py-op">-</tt> <tt class="py-name">x2</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt> <tt class="py-op">-</tt> <tt class="py-name">x3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x2</tt> <tt class="py-op">-</tt> <tt class="py-name">x3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt> <tt class="py-op">-</tt> <tt class="py-name">x4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x2</tt> <tt class="py-op">-</tt> <tt class="py-name">x4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">x3</tt> <tt class="py-op">-</tt> <tt class="py-name">x4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt> <tt class="py-op">-</tt> <tt class="py-name">x5</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x2</tt> <tt class="py-op">-</tt> <tt class="py-name">x5</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x3</tt> <tt class="py-op">-</tt> <tt class="py-name">x5</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x4</tt> <tt class="py-op">-</tt> <tt class="py-name">x5</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> </tt>
</div><a name="L181"></a><tt class="py-lineno">181</tt>  <tt class="py-line"> </tt>
<a name="quintic_extrap"></a><div id="quintic_extrap-def"><a name="L182"></a><tt class="py-lineno">182</tt> <a class="py-toggle" href="#" id="quintic_extrap-toggle" onclick="return toggle('quintic_extrap');">-</a><tt class="py-line"><tt class="py-keyword">def</tt> <a class="py-def-name" href="dadi.Numerics-module.html#quintic_extrap">quintic_extrap</a><tt class="py-op">(</tt><tt class="py-op">(</tt><tt class="py-param">y1</tt><tt class="py-op">,</tt><tt class="py-param">y2</tt><tt class="py-op">,</tt><tt class="py-param">y3</tt><tt class="py-op">,</tt><tt class="py-param">y4</tt><tt class="py-op">,</tt><tt class="py-param">y5</tt><tt class="py-op">,</tt><tt class="py-param">y6</tt><tt class="py-op">)</tt><tt class="py-op">,</tt> <tt class="py-op">(</tt><tt class="py-param">x1</tt><tt class="py-op">,</tt><tt class="py-param">x2</tt><tt class="py-op">,</tt><tt class="py-param">x3</tt><tt class="py-op">,</tt><tt class="py-param">x4</tt><tt class="py-op">,</tt><tt class="py-param">x5</tt><tt class="py-op">,</tt><tt class="py-param">x6</tt><tt class="py-op">)</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
</div><div id="quintic_extrap-collapsed" style="display:none;" pad="+++" indent="++++"></div><div id="quintic_extrap-expanded"><a name="L183"></a><tt class="py-lineno">183</tt>  <tt class="py-line">    <tt class="py-docstring">"""</tt> </tt>
<a name="L184"></a><tt class="py-lineno">184</tt>  <tt class="py-line"><tt class="py-docstring">    Quintic extrapolate from three x,y pairs to x = 0.</tt> </tt>
<a name="L185"></a><tt class="py-lineno">185</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L186"></a><tt class="py-lineno">186</tt>  <tt class="py-line"><tt class="py-docstring">    y1,y2...: y values from x,y pairs. Note that these can be arrays of values.</tt> </tt>
<a name="L187"></a><tt class="py-lineno">187</tt>  <tt class="py-line"><tt class="py-docstring">    x1,x2...: x values from x,y pairs. These should be scalars.</tt> </tt>
<a name="L188"></a><tt class="py-lineno">188</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L189"></a><tt class="py-lineno">189</tt>  <tt class="py-line"><tt class="py-docstring">    Returns extrapolated y at x=0.</tt> </tt>
<a name="L190"></a><tt class="py-lineno">190</tt>  <tt class="py-line"><tt class="py-docstring">    """</tt> </tt>
<a name="L191"></a><tt class="py-lineno">191</tt>  <tt class="py-line">    <tt class="py-comment"># This horrid implementation came from using CForm in Mathematica.</tt> </tt>
<a name="L192"></a><tt class="py-lineno">192</tt>  <tt class="py-line">    <tt class="py-name">Power</tt> <tt class="py-op">=</tt> <tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">power</tt> </tt>
<a name="L193"></a><tt class="py-lineno">193</tt>  <tt class="py-line">    <tt class="py-keyword">return</tt> <tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt> <tt class="py-op">-</tt> <tt class="py-name">x3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">x3</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt> <tt class="py-op">-</tt> <tt class="py-name">x4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x3</tt> <tt class="py-op">-</tt> <tt class="py-name">x4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">x4</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt> <tt class="py-op">-</tt> <tt class="py-name">x5</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">x3</tt> <tt 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class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x2</tt><tt class="py-op">,</tt><tt class="py-number">5</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt> <tt class="py-op">-</tt> <tt class="py-name">x4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">x4</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt> <tt class="py-op">-</tt> <tt class="py-name">x5</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">x4</tt> <tt class="py-op">-</tt> <tt class="py-name">x5</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">x5</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt> <tt class="py-op">-</tt> <tt 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class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">x5</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x6</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y1</tt> <tt class="py-op">-</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y6</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">x4</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">,</tt><tt class="py-number">2</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x6</tt><tt 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class="py-name">y6</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x5</tt><tt class="py-op">,</tt><tt class="py-number">3</tt><tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-op">-</tt><tt class="py-op">(</tt><tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x6</tt><tt class="py-op">,</tt><tt class="py-number">4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y1</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">,</tt><tt class="py-number">4</tt><tt class="py-op">)</tt><tt class="py-op">*</tt><tt class="py-name">y6</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">Power</tt><tt class="py-op">(</tt><tt class="py-name">x4</tt><tt class="py-op">,</tt><tt 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</div><a name="L194"></a><tt class="py-lineno">194</tt>  <tt class="py-line"> </tt>
<a name="reverse_array"></a><div id="reverse_array-def"><a name="L195"></a><tt class="py-lineno">195</tt> <a class="py-toggle" href="#" id="reverse_array-toggle" onclick="return toggle('reverse_array');">-</a><tt class="py-line"><tt class="py-keyword">def</tt> <a class="py-def-name" href="dadi.Numerics-module.html#reverse_array">reverse_array</a><tt class="py-op">(</tt><tt class="py-param">arr</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
</div><div id="reverse_array-collapsed" style="display:none;" pad="+++" indent="++++"></div><div id="reverse_array-expanded"><a name="L196"></a><tt class="py-lineno">196</tt>  <tt class="py-line">    <tt class="py-docstring">"""</tt> </tt>
<a name="L197"></a><tt class="py-lineno">197</tt>  <tt class="py-line"><tt class="py-docstring">    Reverse an array along all axes, so arr[i,j] -&gt; arr[-(i+1),-(j+1)].</tt> </tt>
<a name="L198"></a><tt class="py-lineno">198</tt>  <tt class="py-line"><tt class="py-docstring">    """</tt> </tt>
<a name="L199"></a><tt class="py-lineno">199</tt>  <tt class="py-line">    <tt class="py-name">reverse_slice</tt> <tt class="py-op">=</tt> <tt class="py-op">[</tt><tt class="py-name">slice</tt><tt class="py-op">(</tt><tt class="py-name">None</tt><tt class="py-op">,</tt> <tt class="py-name">None</tt><tt class="py-op">,</tt> <tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">)</tt> <tt class="py-keyword">for</tt> <tt class="py-name">ii</tt> <tt class="py-keyword">in</tt> <tt class="py-name">arr</tt><tt class="py-op">.</tt><tt class="py-name">shape</tt><tt class="py-op">]</tt> </tt>
<a name="L200"></a><tt class="py-lineno">200</tt>  <tt class="py-line">    <tt class="py-keyword">return</tt> <tt class="py-name">arr</tt><tt class="py-op">[</tt><tt class="py-name">reverse_slice</tt><tt class="py-op">]</tt> </tt>
</div><a name="L201"></a><tt class="py-lineno">201</tt>  <tt class="py-line"> </tt>
<a name="intersect_masks"></a><div id="intersect_masks-def"><a name="L202"></a><tt class="py-lineno">202</tt> <a class="py-toggle" href="#" id="intersect_masks-toggle" onclick="return toggle('intersect_masks');">-</a><tt class="py-line"><tt class="py-keyword">def</tt> <a class="py-def-name" href="dadi.Numerics-module.html#intersect_masks">intersect_masks</a><tt class="py-op">(</tt><tt class="py-param">m1</tt><tt class="py-op">,</tt> <tt class="py-param">m2</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
</div><div id="intersect_masks-collapsed" style="display:none;" pad="+++" indent="++++"></div><div id="intersect_masks-expanded"><a name="L203"></a><tt class="py-lineno">203</tt>  <tt class="py-line">    <tt class="py-docstring">"""</tt> </tt>
<a name="L204"></a><tt class="py-lineno">204</tt>  <tt class="py-line"><tt class="py-docstring">    Versions of m1 and m2 that are masked where either m1 or m2 were masked.</tt> </tt>
<a name="L205"></a><tt class="py-lineno">205</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L206"></a><tt class="py-lineno">206</tt>  <tt class="py-line"><tt class="py-docstring">    If neither m1 or m2 is masked, just returns m1 and m2. Otherwise returns</tt> </tt>
<a name="L207"></a><tt class="py-lineno">207</tt>  <tt class="py-line"><tt class="py-docstring">    m1 and m2 wrapped as masked_arrays with identical masks.</tt> </tt>
<a name="L208"></a><tt class="py-lineno">208</tt>  <tt class="py-line"><tt class="py-docstring">    """</tt> </tt>
<a name="L209"></a><tt class="py-lineno">209</tt>  <tt class="py-line">    <tt class="py-name">ma</tt> <tt class="py-op">=</tt> <tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">ma</tt> </tt>
<a name="L210"></a><tt class="py-lineno">210</tt>  <tt class="py-line">    <tt class="py-keyword">import</tt> <tt id="link-2" class="py-name" targets="Package dadi=dadi-module.html"><a title="dadi" class="py-name" href="#" onclick="return doclink('link-2', 'dadi', 'link-2');">dadi</a></tt> </tt>
<a name="L211"></a><tt class="py-lineno">211</tt>  <tt class="py-line">    <tt class="py-keyword">if</tt> <tt class="py-name">ma</tt><tt class="py-op">.</tt><tt class="py-name">isMaskedArray</tt><tt class="py-op">(</tt><tt class="py-name">m1</tt><tt class="py-op">)</tt> <tt class="py-keyword">or</tt> <tt class="py-name">ma</tt><tt class="py-op">.</tt><tt class="py-name">isMaskedArray</tt><tt class="py-op">(</tt><tt class="py-name">m2</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
<a name="L212"></a><tt class="py-lineno">212</tt>  <tt class="py-line">        <tt class="py-name">joint_mask</tt> <tt class="py-op">=</tt> <tt class="py-name">ma</tt><tt class="py-op">.</tt><tt class="py-name">mask_or</tt><tt class="py-op">(</tt><tt class="py-name">ma</tt><tt class="py-op">.</tt><tt class="py-name">getmask</tt><tt class="py-op">(</tt><tt class="py-name">m1</tt><tt class="py-op">)</tt><tt class="py-op">,</tt> <tt class="py-name">ma</tt><tt class="py-op">.</tt><tt class="py-name">getmask</tt><tt class="py-op">(</tt><tt class="py-name">m2</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> </tt>
<a name="L213"></a><tt class="py-lineno">213</tt>  <tt class="py-line"> </tt>
<a name="L214"></a><tt class="py-lineno">214</tt>  <tt class="py-line">        <tt class="py-name">m1</tt> <tt class="py-op">=</tt> <tt id="link-3" class="py-name"><a title="dadi" class="py-name" href="#" onclick="return doclink('link-3', 'dadi', 'link-2');">dadi</a></tt><tt class="py-op">.</tt><tt id="link-4" class="py-name" targets="Variable dadi.Demographics1D.Spectrum=dadi.Spectrum_mod.Spectrum-class.html,Variable dadi.Demographics2D.Spectrum=dadi.Spectrum_mod.Spectrum-class.html,Class dadi.Spectrum_mod.Spectrum=dadi.Spectrum_mod.Spectrum-class.html"><a title="dadi.Demographics1D.Spectrum
dadi.Demographics2D.Spectrum
dadi.Spectrum_mod.Spectrum" class="py-name" href="#" onclick="return doclink('link-4', 'Spectrum', 'link-4');">Spectrum</a></tt><tt class="py-op">(</tt><tt class="py-name">m1</tt><tt class="py-op">,</tt> <tt class="py-name">mask</tt><tt class="py-op">=</tt><tt class="py-name">joint_mask</tt><tt class="py-op">.</tt><tt class="py-name">copy</tt><tt class="py-op">(</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> </tt>
<a name="L215"></a><tt class="py-lineno">215</tt>  <tt class="py-line">        <tt class="py-name">m2</tt> <tt class="py-op">=</tt> <tt id="link-5" class="py-name"><a title="dadi" class="py-name" href="#" onclick="return doclink('link-5', 'dadi', 'link-2');">dadi</a></tt><tt class="py-op">.</tt><tt id="link-6" class="py-name"><a title="dadi.Demographics1D.Spectrum
dadi.Demographics2D.Spectrum
dadi.Spectrum_mod.Spectrum" class="py-name" href="#" onclick="return doclink('link-6', 'Spectrum', 'link-4');">Spectrum</a></tt><tt class="py-op">(</tt><tt class="py-name">m2</tt><tt class="py-op">,</tt> <tt class="py-name">mask</tt><tt class="py-op">=</tt><tt class="py-name">joint_mask</tt><tt class="py-op">.</tt><tt class="py-name">copy</tt><tt class="py-op">(</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> </tt>
<a name="L216"></a><tt class="py-lineno">216</tt>  <tt class="py-line">    <tt class="py-keyword">return</tt> <tt class="py-name">m1</tt><tt class="py-op">,</tt><tt class="py-name">m2</tt> </tt>
</div><a name="L217"></a><tt class="py-lineno">217</tt>  <tt class="py-line"> </tt>
<a name="trapz"></a><div id="trapz-def"><a name="L218"></a><tt class="py-lineno">218</tt> <a class="py-toggle" href="#" id="trapz-toggle" onclick="return toggle('trapz');">-</a><tt class="py-line"><tt class="py-keyword">def</tt> <a class="py-def-name" href="dadi.Numerics-module.html#trapz">trapz</a><tt class="py-op">(</tt><tt class="py-param">yy</tt><tt class="py-op">,</tt> <tt class="py-param">xx</tt><tt class="py-op">=</tt><tt class="py-name">None</tt><tt class="py-op">,</tt> <tt class="py-param">dx</tt><tt class="py-op">=</tt><tt class="py-name">None</tt><tt class="py-op">,</tt> <tt class="py-param">axis</tt><tt class="py-op">=</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
</div><div id="trapz-collapsed" style="display:none;" pad="+++" indent="++++"></div><div id="trapz-expanded"><a name="L219"></a><tt class="py-lineno">219</tt>  <tt class="py-line">    <tt class="py-docstring">"""</tt> </tt>
<a name="L220"></a><tt class="py-lineno">220</tt>  <tt class="py-line"><tt class="py-docstring">    Integrate yy(xx) along given axis using the composite trapezoidal rule.</tt> </tt>
<a name="L221"></a><tt class="py-lineno">221</tt>  <tt class="py-line"><tt class="py-docstring">    </tt> </tt>
<a name="L222"></a><tt class="py-lineno">222</tt>  <tt class="py-line"><tt class="py-docstring">    xx must be one-dimensional and len(xx) must equal yy.shape[axis].</tt> </tt>
<a name="L223"></a><tt class="py-lineno">223</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L224"></a><tt class="py-lineno">224</tt>  <tt class="py-line"><tt class="py-docstring">    This is modified from the SciPy version to work with n-D yy and 1-D xx.</tt> </tt>
<a name="L225"></a><tt class="py-lineno">225</tt>  <tt class="py-line"><tt class="py-docstring">    """</tt> </tt>
<a name="L226"></a><tt class="py-lineno">226</tt>  <tt class="py-line">    <tt class="py-keyword">if</tt> <tt class="py-op">(</tt><tt class="py-name">xx</tt> <tt class="py-keyword">is</tt> <tt class="py-name">None</tt> <tt class="py-keyword">and</tt> <tt class="py-name">dx</tt> <tt class="py-keyword">is</tt> <tt class="py-name">None</tt><tt class="py-op">)</tt>\ </tt>
<a name="L227"></a><tt class="py-lineno">227</tt>  <tt class="py-line">       <tt class="py-keyword">or</tt> <tt class="py-op">(</tt><tt class="py-name">xx</tt> <tt class="py-keyword">is</tt> <tt class="py-keyword">not</tt> <tt class="py-name">None</tt> <tt class="py-keyword">and</tt> <tt class="py-name">dx</tt> <tt class="py-keyword">is</tt> <tt class="py-keyword">not</tt> <tt class="py-name">None</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
<a name="L228"></a><tt class="py-lineno">228</tt>  <tt class="py-line">        <tt class="py-keyword">raise</tt> <tt class="py-name">ValueError</tt><tt class="py-op">(</tt><tt class="py-string">'One and only one of xx or dx must be specified.'</tt><tt class="py-op">)</tt> </tt>
<a name="L229"></a><tt class="py-lineno">229</tt>  <tt class="py-line">    <tt class="py-keyword">elif</tt> <tt class="py-op">(</tt><tt class="py-name">xx</tt> <tt class="py-keyword">is</tt> <tt class="py-keyword">not</tt> <tt class="py-name">None</tt><tt class="py-op">)</tt> <tt class="py-keyword">and</tt> <tt class="py-op">(</tt><tt class="py-name">dx</tt> <tt class="py-keyword">is</tt> <tt class="py-name">None</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
<a name="L230"></a><tt class="py-lineno">230</tt>  <tt class="py-line">        <tt class="py-name">dx</tt> <tt class="py-op">=</tt> <tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">diff</tt><tt class="py-op">(</tt><tt class="py-name">xx</tt><tt class="py-op">)</tt> </tt>
<a name="L231"></a><tt class="py-lineno">231</tt>  <tt class="py-line">    <tt class="py-name">yy</tt> <tt class="py-op">=</tt> <tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">asanyarray</tt><tt class="py-op">(</tt><tt class="py-name">yy</tt><tt class="py-op">)</tt> </tt>
<a name="L232"></a><tt class="py-lineno">232</tt>  <tt class="py-line">    <tt class="py-name">nd</tt> <tt class="py-op">=</tt> <tt class="py-name">yy</tt><tt class="py-op">.</tt><tt class="py-name">ndim</tt> </tt>
<a name="L233"></a><tt class="py-lineno">233</tt>  <tt class="py-line"> </tt>
<a name="L234"></a><tt class="py-lineno">234</tt>  <tt class="py-line">    <tt class="py-keyword">if</tt> <tt class="py-name">yy</tt><tt class="py-op">.</tt><tt class="py-name">shape</tt><tt class="py-op">[</tt><tt class="py-name">axis</tt><tt class="py-op">]</tt> <tt class="py-op">!=</tt> <tt class="py-op">(</tt><tt class="py-name">len</tt><tt class="py-op">(</tt><tt class="py-name">dx</tt><tt class="py-op">)</tt><tt class="py-op">+</tt><tt class="py-number">1</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
<a name="L235"></a><tt class="py-lineno">235</tt>  <tt class="py-line">        <tt class="py-keyword">raise</tt> <tt class="py-name">ValueError</tt><tt class="py-op">(</tt><tt class="py-string">'Length of xx must be equal to length of yy along '</tt> </tt>
<a name="L236"></a><tt class="py-lineno">236</tt>  <tt class="py-line">                         <tt class="py-string">'specified axis. Here len(xx) = %i and '</tt> </tt>
<a name="L237"></a><tt class="py-lineno">237</tt>  <tt class="py-line">                         <tt class="py-string">'yy.shape[axis] = %i.'</tt> <tt class="py-op">%</tt> <tt class="py-op">(</tt><tt class="py-name">len</tt><tt class="py-op">(</tt><tt class="py-name">dx</tt><tt class="py-op">)</tt><tt class="py-op">+</tt><tt class="py-number">1</tt><tt class="py-op">,</tt> <tt class="py-name">yy</tt><tt class="py-op">.</tt><tt class="py-name">shape</tt><tt class="py-op">[</tt><tt class="py-name">axis</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> </tt>
<a name="L238"></a><tt class="py-lineno">238</tt>  <tt class="py-line"> </tt>
<a name="L239"></a><tt class="py-lineno">239</tt>  <tt class="py-line">    <tt class="py-name">slice1</tt> <tt class="py-op">=</tt> <tt class="py-op">[</tt><tt class="py-name">slice</tt><tt class="py-op">(</tt><tt class="py-name">None</tt><tt class="py-op">)</tt><tt class="py-op">]</tt><tt class="py-op">*</tt><tt class="py-name">nd</tt> </tt>
<a name="L240"></a><tt class="py-lineno">240</tt>  <tt class="py-line">    <tt class="py-name">slice2</tt> <tt class="py-op">=</tt> <tt class="py-op">[</tt><tt class="py-name">slice</tt><tt class="py-op">(</tt><tt class="py-name">None</tt><tt class="py-op">)</tt><tt class="py-op">]</tt><tt class="py-op">*</tt><tt class="py-name">nd</tt> </tt>
<a name="L241"></a><tt class="py-lineno">241</tt>  <tt class="py-line">    <tt class="py-name">slice1</tt><tt class="py-op">[</tt><tt class="py-name">axis</tt><tt class="py-op">]</tt> <tt class="py-op">=</tt> <tt class="py-name">slice</tt><tt class="py-op">(</tt><tt class="py-number">1</tt><tt class="py-op">,</tt><tt class="py-name">None</tt><tt class="py-op">)</tt> </tt>
<a name="L242"></a><tt class="py-lineno">242</tt>  <tt class="py-line">    <tt class="py-name">slice2</tt><tt class="py-op">[</tt><tt class="py-name">axis</tt><tt class="py-op">]</tt> <tt class="py-op">=</tt> <tt class="py-name">slice</tt><tt class="py-op">(</tt><tt class="py-name">None</tt><tt class="py-op">,</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">)</tt> </tt>
<a name="L243"></a><tt class="py-lineno">243</tt>  <tt class="py-line">    <tt class="py-name">sliceX</tt> <tt class="py-op">=</tt> <tt class="py-op">[</tt><tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">newaxis</tt><tt class="py-op">]</tt><tt class="py-op">*</tt><tt class="py-name">nd</tt> </tt>
<a name="L244"></a><tt class="py-lineno">244</tt>  <tt class="py-line">    <tt class="py-name">sliceX</tt><tt class="py-op">[</tt><tt class="py-name">axis</tt><tt class="py-op">]</tt> <tt class="py-op">=</tt> <tt class="py-name">slice</tt><tt class="py-op">(</tt><tt class="py-name">None</tt><tt class="py-op">)</tt> </tt>
<a name="L245"></a><tt class="py-lineno">245</tt>  <tt class="py-line"> </tt>
<a name="L246"></a><tt class="py-lineno">246</tt>  <tt class="py-line">    <tt class="py-keyword">return</tt> <tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">sum</tt><tt class="py-op">(</tt><tt class="py-name">dx</tt><tt class="py-op">[</tt><tt class="py-name">sliceX</tt><tt class="py-op">]</tt> <tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">yy</tt><tt class="py-op">[</tt><tt class="py-name">slice1</tt><tt class="py-op">]</tt><tt class="py-op">+</tt><tt class="py-name">yy</tt><tt class="py-op">[</tt><tt class="py-name">slice2</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">/</tt><tt class="py-number">2.0</tt><tt class="py-op">,</tt> <tt class="py-name">axis</tt><tt class="py-op">=</tt><tt class="py-name">axis</tt><tt class="py-op">)</tt> </tt>
</div><a name="L247"></a><tt class="py-lineno">247</tt>  <tt class="py-line"> </tt>
<a name="make_extrap_func"></a><div id="make_extrap_func-def"><a name="L248"></a><tt class="py-lineno">248</tt> <a class="py-toggle" href="#" id="make_extrap_func-toggle" onclick="return toggle('make_extrap_func');">-</a><tt class="py-line"><tt class="py-keyword">def</tt> <a class="py-def-name" href="dadi.Numerics-module.html#make_extrap_func">make_extrap_func</a><tt class="py-op">(</tt><tt class="py-param">func</tt><tt class="py-op">,</tt> <tt class="py-param">extrap_x_l</tt><tt class="py-op">=</tt><tt class="py-name">None</tt><tt class="py-op">,</tt> <tt class="py-param">extrap_log</tt><tt class="py-op">=</tt><tt class="py-name">False</tt><tt class="py-op">,</tt> <tt class="py-param">fail_mag</tt><tt class="py-op">=</tt><tt class="py-number">10</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
</div><div id="make_extrap_func-collapsed" style="display:none;" pad="+++" indent="++++"></div><div id="make_extrap_func-expanded"><a name="L249"></a><tt class="py-lineno">249</tt>  <tt class="py-line">    <tt class="py-docstring">"""</tt> </tt>
<a name="L250"></a><tt class="py-lineno">250</tt>  <tt class="py-line"><tt class="py-docstring">    Generate a version of func that extrapolates to infinitely many gridpoints.</tt> </tt>
<a name="L251"></a><tt class="py-lineno">251</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L252"></a><tt class="py-lineno">252</tt>  <tt class="py-line"><tt class="py-docstring">    func: A function that returns a single scalar or array and whose last</tt> </tt>
<a name="L253"></a><tt class="py-lineno">253</tt>  <tt class="py-line"><tt class="py-docstring">        non-keyword argument is 'pts': the number of default_grid points to use</tt> </tt>
<a name="L254"></a><tt class="py-lineno">254</tt>  <tt class="py-line"><tt class="py-docstring">        in calculation.  </tt> </tt>
<a name="L255"></a><tt class="py-lineno">255</tt>  <tt class="py-line"><tt class="py-docstring">    extrap_x_l: An explict list of x values to use for extrapolation. If not </tt> </tt>
<a name="L256"></a><tt class="py-lineno">256</tt>  <tt class="py-line"><tt class="py-docstring">        provided, the extrapolation routine will look for '.extrap_x'</tt> </tt>
<a name="L257"></a><tt class="py-lineno">257</tt>  <tt class="py-line"><tt class="py-docstring">        attributes on the results of func. The method Spectrum.from_phi will</tt> </tt>
<a name="L258"></a><tt class="py-lineno">258</tt>  <tt class="py-line"><tt class="py-docstring">        add an extrap_x attribute to resulting Spectra, equal to the x-value</tt> </tt>
<a name="L259"></a><tt class="py-lineno">259</tt>  <tt class="py-line"><tt class="py-docstring">        of the first non-zero grid point. An explicit list is useful if you</tt> </tt>
<a name="L260"></a><tt class="py-lineno">260</tt>  <tt class="py-line"><tt class="py-docstring">        want to override this behavior for testing.</tt> </tt>
<a name="L261"></a><tt class="py-lineno">261</tt>  <tt class="py-line"><tt class="py-docstring">    fail_mag:  Simon Gravel noted that there can be numerical instabilities in</tt> </tt>
<a name="L262"></a><tt class="py-lineno">262</tt>  <tt class="py-line"><tt class="py-docstring">        extrapolation when working with large spectra that have very small</tt> </tt>
<a name="L263"></a><tt class="py-lineno">263</tt>  <tt class="py-line"><tt class="py-docstring">        entires (of order 1e-24). To avoid these instabilities, we ignore the </tt> </tt>
<a name="L264"></a><tt class="py-lineno">264</tt>  <tt class="py-line"><tt class="py-docstring">        extrapolation values (and use the input result with the smallest x) </tt> </tt>
<a name="L265"></a><tt class="py-lineno">265</tt>  <tt class="py-line"><tt class="py-docstring">        if the extrapolation is more than fail_mag orders of magnitude away</tt> </tt>
<a name="L266"></a><tt class="py-lineno">266</tt>  <tt class="py-line"><tt class="py-docstring">        from the smallest x input result.</tt> </tt>
<a name="L267"></a><tt class="py-lineno">267</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L268"></a><tt class="py-lineno">268</tt>  <tt class="py-line"><tt class="py-docstring">    Returns a new function whose last argument is a list of numbers of grid</tt> </tt>
<a name="L269"></a><tt class="py-lineno">269</tt>  <tt class="py-line"><tt class="py-docstring">    points and that returns a result extrapolated to infinitely many grid</tt> </tt>
<a name="L270"></a><tt class="py-lineno">270</tt>  <tt class="py-line"><tt class="py-docstring">    points.</tt> </tt>
<a name="L271"></a><tt class="py-lineno">271</tt>  <tt class="py-line"><tt class="py-docstring">    """</tt> </tt>
<a name="L272"></a><tt class="py-lineno">272</tt>  <tt class="py-line">    <tt class="py-name">x_l_from_results</tt> <tt class="py-op">=</tt> <tt class="py-op">(</tt><tt class="py-name">extrap_x_l</tt> <tt class="py-keyword">is</tt> <tt class="py-name">None</tt><tt class="py-op">)</tt> </tt>
<a name="L273"></a><tt class="py-lineno">273</tt>  <tt class="py-line"> </tt>
<a name="L274"></a><tt class="py-lineno">274</tt>  <tt class="py-line">    <tt class="py-keyword">def</tt> <tt class="py-def-name">extrap_func</tt><tt class="py-op">(</tt><tt class="py-op">*</tt><tt class="py-param">args</tt><tt class="py-op">,</tt> <tt class="py-op">**</tt><tt class="py-param">kwargs</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
<a name="L275"></a><tt class="py-lineno">275</tt>  <tt class="py-line">        <tt class="py-comment"># Separate pts (or pts_l) from arguments</tt> </tt>
<a name="L276"></a><tt class="py-lineno">276</tt>  <tt class="py-line">        <tt class="py-keyword">if</tt> <tt class="py-string">'pts'</tt> <tt class="py-keyword">not</tt> <tt class="py-keyword">in</tt> <tt class="py-name">kwargs</tt><tt class="py-op">:</tt> </tt>
<a name="L277"></a><tt class="py-lineno">277</tt>  <tt class="py-line">            <tt class="py-name">other_args</tt><tt class="py-op">,</tt> <tt class="py-name">pts_l</tt> <tt class="py-op">=</tt> <tt class="py-name">args</tt><tt class="py-op">[</tt><tt class="py-op">:</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt><tt class="py-op">,</tt> <tt class="py-name">args</tt><tt class="py-op">[</tt><tt class="py-op">-</tt><tt class="py-number">1</tt><tt class="py-op">]</tt> </tt>
<a name="L278"></a><tt class="py-lineno">278</tt>  <tt class="py-line">        <tt class="py-keyword">else</tt><tt class="py-op">:</tt> </tt>
<a name="L279"></a><tt class="py-lineno">279</tt>  <tt class="py-line">            <tt class="py-name">other_args</tt> <tt class="py-op">=</tt> <tt class="py-name">args</tt> </tt>
<a name="L280"></a><tt class="py-lineno">280</tt>  <tt class="py-line">            <tt class="py-name">pts_l</tt> <tt class="py-op">=</tt> <tt class="py-name">kwargs</tt><tt class="py-op">[</tt><tt class="py-string">'pts'</tt><tt class="py-op">]</tt> </tt>
<a name="L281"></a><tt class="py-lineno">281</tt>  <tt class="py-line">            <tt class="py-keyword">del</tt> <tt class="py-name">kwargs</tt><tt class="py-op">[</tt><tt class="py-string">'pts'</tt><tt class="py-op">]</tt> </tt>
<a name="L282"></a><tt class="py-lineno">282</tt>  <tt class="py-line"> </tt>
<a name="L283"></a><tt class="py-lineno">283</tt>  <tt class="py-line">        <tt class="py-keyword">if</tt> <tt class="py-string">'no_extrap'</tt> <tt class="py-keyword">in</tt> <tt class="py-name">kwargs</tt><tt class="py-op">:</tt> </tt>
<a name="L284"></a><tt class="py-lineno">284</tt>  <tt class="py-line">            <tt class="py-name">no_extrap</tt> <tt class="py-op">=</tt> <tt class="py-name">True</tt> </tt>
<a name="L285"></a><tt class="py-lineno">285</tt>  <tt class="py-line">            <tt class="py-keyword">del</tt> <tt class="py-name">kwargs</tt><tt class="py-op">[</tt><tt class="py-string">'no_extrap'</tt><tt class="py-op">]</tt> </tt>
<a name="L286"></a><tt class="py-lineno">286</tt>  <tt class="py-line">        <tt class="py-keyword">else</tt><tt class="py-op">:</tt> </tt>
<a name="L287"></a><tt class="py-lineno">287</tt>  <tt class="py-line">            <tt class="py-name">no_extrap</tt> <tt class="py-op">=</tt> <tt class="py-name">False</tt> </tt>
<a name="L288"></a><tt class="py-lineno">288</tt>  <tt class="py-line"> </tt>
<a name="L289"></a><tt class="py-lineno">289</tt>  <tt class="py-line">        <tt class="py-keyword">if</tt> <tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">isscalar</tt><tt class="py-op">(</tt><tt class="py-name">pts_l</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
<a name="L290"></a><tt class="py-lineno">290</tt>  <tt class="py-line">            <tt class="py-name">pts_l</tt> <tt class="py-op">=</tt> <tt class="py-op">[</tt><tt class="py-name">pts_l</tt><tt class="py-op">]</tt> </tt>
<a name="L291"></a><tt class="py-lineno">291</tt>  <tt class="py-line"> </tt>
<a name="L292"></a><tt class="py-lineno">292</tt>  <tt class="py-line">        <tt class="py-comment"># Create a sub-function that fixes all other arguments and only</tt> </tt>
<a name="L293"></a><tt class="py-lineno">293</tt>  <tt class="py-line">        <tt class="py-comment"># takes in pts.</tt> </tt>
<a name="L294"></a><tt class="py-lineno">294</tt>  <tt class="py-line">        <tt class="py-name">partial_func</tt> <tt class="py-op">=</tt> <tt class="py-name">functools</tt><tt class="py-op">.</tt><tt class="py-name">partial</tt><tt class="py-op">(</tt><tt class="py-name">func</tt><tt class="py-op">,</tt> <tt class="py-op">*</tt><tt class="py-name">other_args</tt><tt class="py-op">,</tt> <tt class="py-op">**</tt><tt class="py-name">kwargs</tt><tt class="py-op">)</tt> </tt>
<a name="L295"></a><tt class="py-lineno">295</tt>  <tt class="py-line"> </tt>
<a name="L296"></a><tt class="py-lineno">296</tt>  <tt class="py-line">        <tt class="py-comment">##</tt> </tt>
<a name="L297"></a><tt class="py-lineno">297</tt>  <tt class="py-line">        <tt class="py-comment">## The commented-out code implements distribution of fs calculations</tt> </tt>
<a name="L298"></a><tt class="py-lineno">298</tt>  <tt class="py-line">        <tt class="py-comment">## among multiple processors. Unfortunately, it doesn't work with</tt> </tt>
<a name="L299"></a><tt class="py-lineno">299</tt>  <tt class="py-line">        <tt class="py-comment">## iPython, because pickling functions is very fragile in the</tt> </tt>
<a name="L300"></a><tt class="py-lineno">300</tt>  <tt class="py-line">        <tt class="py-comment">## interactive interpreter. If we had some sort of data structure for</tt> </tt>
<a name="L301"></a><tt class="py-lineno">301</tt>  <tt class="py-line">        <tt class="py-comment">## defining models, then this would be much easier to implement. Note</tt> </tt>
<a name="L302"></a><tt class="py-lineno">302</tt>  <tt class="py-line">        <tt class="py-comment">## that there might still be issues on Windows systems, because of its</tt> </tt>
<a name="L303"></a><tt class="py-lineno">303</tt>  <tt class="py-line">        <tt class="py-comment">## poor-man's version of fork().</tt> </tt>
<a name="L304"></a><tt class="py-lineno">304</tt>  <tt class="py-line">        <tt class="py-comment">##</tt> </tt>
<a name="L305"></a><tt class="py-lineno">305</tt>  <tt class="py-line">        <tt class="py-comment">#import cPickle, multiprocessing</tt> </tt>
<a name="L306"></a><tt class="py-lineno">306</tt>  <tt class="py-line">        <tt class="py-comment">#try:</tt> </tt>
<a name="L307"></a><tt class="py-lineno">307</tt>  <tt class="py-line">        <tt class="py-comment">#    # Test whether sub-function is picklable. If not, pool.map will</tt> </tt>
<a name="L308"></a><tt class="py-lineno">308</tt>  <tt class="py-line">        <tt class="py-comment">#    # hang.</tt> </tt>
<a name="L309"></a><tt class="py-lineno">309</tt>  <tt class="py-line">        <tt class="py-comment">#    cPickle.dumps(partial_func)</tt> </tt>
<a name="L310"></a><tt class="py-lineno">310</tt>  <tt class="py-line">        <tt class="py-comment">#    pool = multiprocessing.Pool()</tt> </tt>
<a name="L311"></a><tt class="py-lineno">311</tt>  <tt class="py-line">        <tt class="py-comment">#    result_l = pool.map(partial_func, pts_l)</tt> </tt>
<a name="L312"></a><tt class="py-lineno">312</tt>  <tt class="py-line">        <tt class="py-comment">#    pool.close()</tt> </tt>
<a name="L313"></a><tt class="py-lineno">313</tt>  <tt class="py-line">        <tt class="py-comment">#except (cPickle.PicklingError, TypeError):</tt> </tt>
<a name="L314"></a><tt class="py-lineno">314</tt>  <tt class="py-line">        <tt class="py-comment">#    print('Function passed to extrap func must be picklable for '</tt> </tt>
<a name="L315"></a><tt class="py-lineno">315</tt>  <tt class="py-line">        <tt class="py-comment">#          'multi-processor executation.')</tt> </tt>
<a name="L316"></a><tt class="py-lineno">316</tt>  <tt class="py-line">        <tt class="py-comment">#    import sys</tt> </tt>
<a name="L317"></a><tt class="py-lineno">317</tt>  <tt class="py-line">        <tt class="py-comment">#    sys.exit()</tt> </tt>
<a name="L318"></a><tt class="py-lineno">318</tt>  <tt class="py-line"> </tt>
<a name="L319"></a><tt class="py-lineno">319</tt>  <tt class="py-line">        <tt class="py-name">result_l</tt> <tt class="py-op">=</tt> <tt class="py-name">map</tt><tt class="py-op">(</tt><tt class="py-name">partial_func</tt><tt class="py-op">,</tt> <tt class="py-name">pts_l</tt><tt class="py-op">)</tt> </tt>
<a name="L320"></a><tt class="py-lineno">320</tt>  <tt class="py-line">        <tt class="py-keyword">if</tt> <tt class="py-name">no_extrap</tt><tt class="py-op">:</tt> </tt>
<a name="L321"></a><tt class="py-lineno">321</tt>  <tt class="py-line">            <tt class="py-keyword">return</tt> <tt class="py-name">result_l</tt> </tt>
<a name="L322"></a><tt class="py-lineno">322</tt>  <tt class="py-line"> </tt>
<a name="L323"></a><tt class="py-lineno">323</tt>  <tt class="py-line">        <tt class="py-keyword">if</tt> <tt class="py-name">x_l_from_results</tt><tt class="py-op">:</tt> </tt>
<a name="L324"></a><tt class="py-lineno">324</tt>  <tt class="py-line">            <tt class="py-keyword">try</tt><tt class="py-op">:</tt> </tt>
<a name="L325"></a><tt class="py-lineno">325</tt>  <tt class="py-line">                <tt class="py-name">x_l</tt> <tt class="py-op">=</tt> <tt class="py-op">[</tt><tt class="py-name">r</tt><tt class="py-op">.</tt><tt class="py-name">extrap_x</tt> <tt class="py-keyword">for</tt> <tt class="py-name">r</tt> <tt class="py-keyword">in</tt> <tt class="py-name">result_l</tt><tt class="py-op">]</tt> </tt>
<a name="L326"></a><tt class="py-lineno">326</tt>  <tt class="py-line">            <tt class="py-keyword">except</tt> <tt class="py-name">AttributeError</tt><tt class="py-op">:</tt> </tt>
<a name="L327"></a><tt class="py-lineno">327</tt>  <tt class="py-line">                <tt class="py-keyword">raise</tt> <tt class="py-name">ValueError</tt><tt class="py-op">(</tt><tt class="py-string">"Extrapolation function error: No explicit extrapolation x_l provided, and results do not have 'extrap_x' attributes. If this is an FS extrapolation, check your from_phi method."</tt><tt class="py-op">)</tt> </tt>
<a name="L328"></a><tt class="py-lineno">328</tt>  <tt class="py-line">        <tt class="py-keyword">else</tt><tt class="py-op">:</tt> </tt>
<a name="L329"></a><tt class="py-lineno">329</tt>  <tt class="py-line">            <tt class="py-name">x_l</tt> <tt class="py-op">=</tt> <tt class="py-name">extrap_x_l</tt> </tt>
<a name="L330"></a><tt class="py-lineno">330</tt>  <tt class="py-line"> </tt>
<a name="L331"></a><tt class="py-lineno">331</tt>  <tt class="py-line">        <tt class="py-keyword">if</tt> <tt class="py-name">extrap_log</tt><tt class="py-op">:</tt> </tt>
<a name="L332"></a><tt class="py-lineno">332</tt>  <tt class="py-line">            <tt class="py-name">result_l</tt> <tt class="py-op">=</tt> <tt class="py-op">[</tt><tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">log</tt><tt class="py-op">(</tt><tt class="py-name">r</tt><tt class="py-op">)</tt> <tt class="py-keyword">for</tt> <tt class="py-name">r</tt> <tt class="py-keyword">in</tt> <tt class="py-name">result_l</tt><tt class="py-op">]</tt> </tt>
<a name="L333"></a><tt class="py-lineno">333</tt>  <tt class="py-line"> </tt>
<a name="L334"></a><tt class="py-lineno">334</tt>  <tt class="py-line">        <tt class="py-comment"># Extrapolate</tt> </tt>
<a name="L335"></a><tt class="py-lineno">335</tt>  <tt class="py-line">        <tt class="py-keyword">if</tt> <tt class="py-name">len</tt><tt class="py-op">(</tt><tt class="py-name">pts_l</tt><tt class="py-op">)</tt> <tt class="py-op">==</tt> <tt class="py-number">1</tt><tt class="py-op">:</tt> </tt>
<a name="L336"></a><tt class="py-lineno">336</tt>  <tt class="py-line">            <tt class="py-name">ex_result</tt> <tt class="py-op">=</tt> <tt class="py-name">result_l</tt><tt class="py-op">[</tt><tt class="py-number">0</tt><tt class="py-op">]</tt> </tt>
<a name="L337"></a><tt class="py-lineno">337</tt>  <tt class="py-line">        <tt class="py-keyword">elif</tt> <tt class="py-name">len</tt><tt class="py-op">(</tt><tt class="py-name">pts_l</tt><tt class="py-op">)</tt> <tt class="py-op">==</tt> <tt class="py-number">2</tt><tt class="py-op">:</tt> </tt>
<a name="L338"></a><tt class="py-lineno">338</tt>  <tt class="py-line">            <tt class="py-name">ex_result</tt> <tt class="py-op">=</tt> <tt id="link-7" class="py-name" targets="Function dadi.Numerics.linear_extrap()=dadi.Numerics-module.html#linear_extrap"><a title="dadi.Numerics.linear_extrap" class="py-name" href="#" onclick="return doclink('link-7', 'linear_extrap', 'link-7');">linear_extrap</a></tt><tt class="py-op">(</tt><tt class="py-name">result_l</tt><tt class="py-op">,</tt> <tt class="py-name">x_l</tt><tt class="py-op">)</tt> </tt>
<a name="L339"></a><tt class="py-lineno">339</tt>  <tt class="py-line">        <tt class="py-keyword">elif</tt> <tt class="py-name">len</tt><tt class="py-op">(</tt><tt class="py-name">pts_l</tt><tt class="py-op">)</tt> <tt class="py-op">==</tt> <tt class="py-number">3</tt><tt class="py-op">:</tt> </tt>
<a name="L340"></a><tt class="py-lineno">340</tt>  <tt class="py-line">            <tt class="py-name">ex_result</tt> <tt class="py-op">=</tt> <tt id="link-8" class="py-name" targets="Function dadi.Numerics.quadratic_extrap()=dadi.Numerics-module.html#quadratic_extrap"><a title="dadi.Numerics.quadratic_extrap" class="py-name" href="#" onclick="return doclink('link-8', 'quadratic_extrap', 'link-8');">quadratic_extrap</a></tt><tt class="py-op">(</tt><tt class="py-name">result_l</tt><tt class="py-op">,</tt> <tt class="py-name">x_l</tt><tt class="py-op">)</tt> </tt>
<a name="L341"></a><tt class="py-lineno">341</tt>  <tt class="py-line">        <tt class="py-keyword">elif</tt> <tt class="py-name">len</tt><tt class="py-op">(</tt><tt class="py-name">pts_l</tt><tt class="py-op">)</tt> <tt class="py-op">==</tt> <tt class="py-number">4</tt><tt class="py-op">:</tt> </tt>
<a name="L342"></a><tt class="py-lineno">342</tt>  <tt class="py-line">            <tt class="py-name">ex_result</tt> <tt class="py-op">=</tt> <tt id="link-9" class="py-name" targets="Function dadi.Numerics.cubic_extrap()=dadi.Numerics-module.html#cubic_extrap"><a title="dadi.Numerics.cubic_extrap" class="py-name" href="#" onclick="return doclink('link-9', 'cubic_extrap', 'link-9');">cubic_extrap</a></tt><tt class="py-op">(</tt><tt class="py-name">result_l</tt><tt class="py-op">,</tt> <tt class="py-name">x_l</tt><tt class="py-op">)</tt> </tt>
<a name="L343"></a><tt class="py-lineno">343</tt>  <tt class="py-line">        <tt class="py-keyword">elif</tt> <tt class="py-name">len</tt><tt class="py-op">(</tt><tt class="py-name">pts_l</tt><tt class="py-op">)</tt> <tt class="py-op">==</tt> <tt class="py-number">5</tt><tt class="py-op">:</tt> </tt>
<a name="L344"></a><tt class="py-lineno">344</tt>  <tt class="py-line">            <tt class="py-name">ex_result</tt> <tt class="py-op">=</tt> <tt id="link-10" class="py-name" targets="Function dadi.Numerics.quartic_extrap()=dadi.Numerics-module.html#quartic_extrap"><a title="dadi.Numerics.quartic_extrap" class="py-name" href="#" onclick="return doclink('link-10', 'quartic_extrap', 'link-10');">quartic_extrap</a></tt><tt class="py-op">(</tt><tt class="py-name">result_l</tt><tt class="py-op">,</tt> <tt class="py-name">x_l</tt><tt class="py-op">)</tt> </tt>
<a name="L345"></a><tt class="py-lineno">345</tt>  <tt class="py-line">        <tt class="py-keyword">elif</tt> <tt class="py-name">len</tt><tt class="py-op">(</tt><tt class="py-name">pts_l</tt><tt class="py-op">)</tt> <tt class="py-op">==</tt> <tt class="py-number">6</tt><tt class="py-op">:</tt> </tt>
<a name="L346"></a><tt class="py-lineno">346</tt>  <tt class="py-line">            <tt class="py-name">ex_result</tt> <tt class="py-op">=</tt> <tt id="link-11" class="py-name" targets="Function dadi.Numerics.quintic_extrap()=dadi.Numerics-module.html#quintic_extrap"><a title="dadi.Numerics.quintic_extrap" class="py-name" href="#" onclick="return doclink('link-11', 'quintic_extrap', 'link-11');">quintic_extrap</a></tt><tt class="py-op">(</tt><tt class="py-name">result_l</tt><tt class="py-op">,</tt> <tt class="py-name">x_l</tt><tt class="py-op">)</tt> </tt>
<a name="L347"></a><tt class="py-lineno">347</tt>  <tt class="py-line">        <tt class="py-keyword">else</tt><tt class="py-op">:</tt> </tt>
<a name="L348"></a><tt class="py-lineno">348</tt>  <tt class="py-line">            <tt class="py-keyword">raise</tt> <tt class="py-name">ValueError</tt><tt class="py-op">(</tt><tt class="py-string">'Number of calculations to use for extrapolation '</tt> </tt>
<a name="L349"></a><tt class="py-lineno">349</tt>  <tt class="py-line">                             <tt class="py-string">'must be between 1 and 6'</tt><tt class="py-op">)</tt> </tt>
<a name="L350"></a><tt class="py-lineno">350</tt>  <tt class="py-line"> </tt>
<a name="L351"></a><tt class="py-lineno">351</tt>  <tt class="py-line">        <tt class="py-keyword">if</tt> <tt class="py-name">extrap_log</tt><tt class="py-op">:</tt> </tt>
<a name="L352"></a><tt class="py-lineno">352</tt>  <tt class="py-line">            <tt class="py-name">ex_result</tt> <tt class="py-op">=</tt> <tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">exp</tt><tt class="py-op">(</tt><tt class="py-name">ex_result</tt><tt class="py-op">)</tt> </tt>
<a name="L353"></a><tt class="py-lineno">353</tt>  <tt class="py-line"> </tt>
<a name="L354"></a><tt class="py-lineno">354</tt>  <tt class="py-line">        <tt class="py-comment"># Simon Gravel noted that there can be numerical instabilities in</tt> </tt>
<a name="L355"></a><tt class="py-lineno">355</tt>  <tt class="py-line">        <tt class="py-comment"># extrapolation when working with large spectra that have very small</tt> </tt>
<a name="L356"></a><tt class="py-lineno">356</tt>  <tt class="py-line">        <tt class="py-comment"># entires (of order 1e-24).</tt> </tt>
<a name="L357"></a><tt class="py-lineno">357</tt>  <tt class="py-line">        <tt class="py-comment"># To avoid these instabilities, we ignore the extrapolation values</tt> </tt>
<a name="L358"></a><tt class="py-lineno">358</tt>  <tt class="py-line">        <tt class="py-comment"># if it is too different from the input values.</tt> </tt>
<a name="L359"></a><tt class="py-lineno">359</tt>  <tt class="py-line">        <tt class="py-keyword">if</tt> <tt class="py-name">len</tt><tt class="py-op">(</tt><tt class="py-name">pts_l</tt><tt class="py-op">)</tt> <tt class="py-op">&gt;</tt> <tt class="py-number">1</tt><tt class="py-op">:</tt> </tt>
<a name="L360"></a><tt class="py-lineno">360</tt>  <tt class="py-line">            <tt class="py-comment"># Assume the best input value comes from the smallest grid.</tt> </tt>
<a name="L361"></a><tt class="py-lineno">361</tt>  <tt class="py-line">            <tt class="py-name">best_result</tt> <tt class="py-op">=</tt> <tt class="py-name">result_l</tt><tt class="py-op">[</tt><tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">argmin</tt><tt class="py-op">(</tt><tt class="py-name">x_l</tt><tt class="py-op">)</tt><tt class="py-op">]</tt> </tt>
<a name="L362"></a><tt class="py-lineno">362</tt>  <tt class="py-line">            <tt class="py-keyword">if</tt> <tt class="py-name">extrap_log</tt><tt class="py-op">:</tt> </tt>
<a name="L363"></a><tt class="py-lineno">363</tt>  <tt class="py-line">                <tt class="py-name">best_result</tt> <tt class="py-op">=</tt> <tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">exp</tt><tt class="py-op">(</tt><tt class="py-name">best_result</tt><tt class="py-op">)</tt> </tt>
<a name="L364"></a><tt class="py-lineno">364</tt>  <tt class="py-line"> </tt>
<a name="L365"></a><tt class="py-lineno">365</tt>  <tt class="py-line">            <tt class="py-comment"># The extrapolation is deemed to have failed if it results in a</tt> </tt>
<a name="L366"></a><tt class="py-lineno">366</tt>  <tt class="py-line">            <tt class="py-comment"># value more than fail_mag orders of magnitude away from the </tt> </tt>
<a name="L367"></a><tt class="py-lineno">367</tt>  <tt class="py-line">            <tt class="py-comment"># best input value.</tt> </tt>
<a name="L368"></a><tt class="py-lineno">368</tt>  <tt class="py-line">            <tt class="py-name">extrap_failed</tt> <tt class="py-op">=</tt> <tt class="py-name">abs</tt><tt class="py-op">(</tt><tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">log10</tt><tt class="py-op">(</tt><tt class="py-name">ex_result</tt><tt class="py-op">/</tt><tt class="py-name">best_result</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> <tt class="py-op">&gt;</tt> <tt class="py-name">fail_mag</tt> </tt>
<a name="L369"></a><tt class="py-lineno">369</tt>  <tt class="py-line">            <tt class="py-keyword">if</tt> <tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">any</tt><tt class="py-op">(</tt><tt class="py-name">extrap_failed</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
<a name="L370"></a><tt class="py-lineno">370</tt>  <tt class="py-line">                <tt id="link-12" class="py-name"><a title="dadi.Hessian.logger
dadi.Inference.logger
dadi.Integration.logger
dadi.Numerics.logger
dadi.RunInParallel.logger
dadi.Spectrum_mod.logger" class="py-name" href="#" onclick="return doclink('link-12', 'logger', 'link-0');">logger</a></tt><tt class="py-op">.</tt><tt class="py-name">warn</tt><tt class="py-op">(</tt><tt class="py-string">'Extrapolation may have failed. Check resulting '</tt> </tt>
<a name="L371"></a><tt class="py-lineno">371</tt>  <tt class="py-line">                            <tt class="py-string">'frequency spectrum for unexpected results.'</tt><tt class="py-op">)</tt> </tt>
<a name="L372"></a><tt class="py-lineno">372</tt>  <tt class="py-line"> </tt>
<a name="L373"></a><tt class="py-lineno">373</tt>  <tt class="py-line">            <tt class="py-comment"># For entries that fail, use the "best" input result.</tt> </tt>
<a name="L374"></a><tt class="py-lineno">374</tt>  <tt class="py-line">            <tt class="py-name">ex_result</tt><tt class="py-op">[</tt><tt class="py-name">extrap_failed</tt><tt class="py-op">]</tt> <tt class="py-op">=</tt> <tt class="py-name">best_result</tt><tt class="py-op">[</tt><tt class="py-name">extrap_failed</tt><tt class="py-op">]</tt> </tt>
<a name="L375"></a><tt class="py-lineno">375</tt>  <tt class="py-line"> </tt>
<a name="L376"></a><tt class="py-lineno">376</tt>  <tt class="py-line">        <tt class="py-keyword">return</tt> <tt class="py-name">ex_result</tt> </tt>
</div><a name="L377"></a><tt class="py-lineno">377</tt>  <tt class="py-line"> </tt>
<a name="L378"></a><tt class="py-lineno">378</tt>  <tt class="py-line">    <tt class="py-name">extrap_func</tt><tt class="py-op">.</tt><tt class="py-name">func_name</tt> <tt class="py-op">=</tt> <tt class="py-name">func</tt><tt class="py-op">.</tt><tt class="py-name">func_name</tt> </tt>
<a name="L379"></a><tt class="py-lineno">379</tt>  <tt class="py-line">    <tt class="py-name">extrap_func</tt><tt class="py-op">.</tt><tt class="py-name">func_doc</tt> <tt class="py-op">=</tt> <tt class="py-name">func</tt><tt class="py-op">.</tt><tt class="py-name">func_doc</tt> </tt>
<a name="L380"></a><tt class="py-lineno">380</tt>  <tt class="py-line"> </tt>
<a name="L381"></a><tt class="py-lineno">381</tt>  <tt class="py-line">    <tt class="py-keyword">return</tt> <tt class="py-name">extrap_func</tt> </tt>
</div><a name="L382"></a><tt class="py-lineno">382</tt>  <tt class="py-line"> </tt>
<a name="make_extrap_log_func"></a><div id="make_extrap_log_func-def"><a name="L383"></a><tt class="py-lineno">383</tt> <a class="py-toggle" href="#" id="make_extrap_log_func-toggle" onclick="return toggle('make_extrap_log_func');">-</a><tt class="py-line"><tt class="py-keyword">def</tt> <a class="py-def-name" href="dadi.Numerics-module.html#make_extrap_log_func">make_extrap_log_func</a><tt class="py-op">(</tt><tt class="py-param">func</tt><tt class="py-op">,</tt> <tt class="py-param">extrap_x_l</tt><tt class="py-op">=</tt><tt class="py-name">None</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
</div><div id="make_extrap_log_func-collapsed" style="display:none;" pad="+++" indent="++++"></div><div id="make_extrap_log_func-expanded"><a name="L384"></a><tt class="py-lineno">384</tt>  <tt class="py-line">    <tt class="py-docstring">"""</tt> </tt>
<a name="L385"></a><tt class="py-lineno">385</tt>  <tt class="py-line"><tt class="py-docstring">    Generate a version of func that extrapolates to infinitely many gridpoints.</tt> </tt>
<a name="L386"></a><tt class="py-lineno">386</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L387"></a><tt class="py-lineno">387</tt>  <tt class="py-line"><tt class="py-docstring">    Note that extrapolation here is done on the *log* of the function result,</tt> </tt>
<a name="L388"></a><tt class="py-lineno">388</tt>  <tt class="py-line"><tt class="py-docstring">    so this will fail if any returned values are &lt; 0. It does seem to be better</tt> </tt>
<a name="L389"></a><tt class="py-lineno">389</tt>  <tt class="py-line"><tt class="py-docstring">    behaved for SFS calculation.</tt> </tt>
<a name="L390"></a><tt class="py-lineno">390</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L391"></a><tt class="py-lineno">391</tt>  <tt class="py-line"><tt class="py-docstring">    func: A function whose last argument is the number of Numerics.default_grid </tt> </tt>
<a name="L392"></a><tt class="py-lineno">392</tt>  <tt class="py-line"><tt class="py-docstring">          points to use in calculation and that returns a single scalar or </tt> </tt>
<a name="L393"></a><tt class="py-lineno">393</tt>  <tt class="py-line"><tt class="py-docstring">          array.</tt> </tt>
<a name="L394"></a><tt class="py-lineno">394</tt>  <tt class="py-line"><tt class="py-docstring">    extrap_x_l: An explict list of x values to use for extrapolation. If not </tt> </tt>
<a name="L395"></a><tt class="py-lineno">395</tt>  <tt class="py-line"><tt class="py-docstring">         provided, the extrapolation routine will look for '.extrap_x'</tt> </tt>
<a name="L396"></a><tt class="py-lineno">396</tt>  <tt class="py-line"><tt class="py-docstring">         attributes on the results of func. The method Spectrum.from_phi will</tt> </tt>
<a name="L397"></a><tt class="py-lineno">397</tt>  <tt class="py-line"><tt class="py-docstring">         add an extrap_x attribute to resulting Spectra, equal to the x-value</tt> </tt>
<a name="L398"></a><tt class="py-lineno">398</tt>  <tt class="py-line"><tt class="py-docstring">         of the first non-zero grid point. An explicit list is useful if you</tt> </tt>
<a name="L399"></a><tt class="py-lineno">399</tt>  <tt class="py-line"><tt class="py-docstring">         want to override this behavior for testing.</tt> </tt>
<a name="L400"></a><tt class="py-lineno">400</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L401"></a><tt class="py-lineno">401</tt>  <tt class="py-line"><tt class="py-docstring">    Returns a new function whose last argument is a list of numbers of grid</tt> </tt>
<a name="L402"></a><tt class="py-lineno">402</tt>  <tt class="py-line"><tt class="py-docstring">    points and that returns a result extrapolated to infinitely many grid</tt> </tt>
<a name="L403"></a><tt class="py-lineno">403</tt>  <tt class="py-line"><tt class="py-docstring">    points.</tt> </tt>
<a name="L404"></a><tt class="py-lineno">404</tt>  <tt class="py-line"><tt class="py-docstring">    """</tt> </tt>
<a name="L405"></a><tt class="py-lineno">405</tt>  <tt class="py-line">    <tt class="py-keyword">return</tt> <tt id="link-13" class="py-name" targets="Function dadi.Numerics.make_extrap_func()=dadi.Numerics-module.html#make_extrap_func"><a title="dadi.Numerics.make_extrap_func" class="py-name" href="#" onclick="return doclink('link-13', 'make_extrap_func', 'link-13');">make_extrap_func</a></tt><tt class="py-op">(</tt><tt class="py-name">func</tt><tt class="py-op">,</tt> <tt class="py-name">extrap_x_l</tt><tt class="py-op">=</tt><tt class="py-name">extrap_x_l</tt><tt class="py-op">,</tt> <tt class="py-name">extrap_log</tt><tt class="py-op">=</tt><tt class="py-name">True</tt><tt class="py-op">)</tt> </tt>
</div><a name="L406"></a><tt class="py-lineno">406</tt>  <tt class="py-line"> </tt>
<a name="L407"></a><tt class="py-lineno">407</tt>  <tt class="py-line"><tt id="link-14" class="py-name" targets="Variable dadi.Numerics._projection_cache=dadi.Numerics-module.html#_projection_cache"><a title="dadi.Numerics._projection_cache" class="py-name" href="#" onclick="return doclink('link-14', '_projection_cache', 'link-14');">_projection_cache</a></tt> <tt class="py-op">=</tt> <tt class="py-op">{</tt><tt class="py-op">}</tt> </tt>
<a name="_lncomb"></a><div id="_lncomb-def"><a name="L408"></a><tt class="py-lineno">408</tt> <a class="py-toggle" href="#" id="_lncomb-toggle" onclick="return toggle('_lncomb');">-</a><tt class="py-line"><tt class="py-keyword">def</tt> <a class="py-def-name" href="dadi.Numerics-module.html#_lncomb">_lncomb</a><tt class="py-op">(</tt><tt class="py-param">N</tt><tt class="py-op">,</tt><tt class="py-param">k</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
</div><div id="_lncomb-collapsed" style="display:none;" pad="+++" indent="++++"></div><div id="_lncomb-expanded"><a name="L409"></a><tt class="py-lineno">409</tt>  <tt class="py-line">    <tt class="py-docstring">"""</tt> </tt>
<a name="L410"></a><tt class="py-lineno">410</tt>  <tt class="py-line"><tt class="py-docstring">    Log of N choose k.</tt> </tt>
<a name="L411"></a><tt class="py-lineno">411</tt>  <tt class="py-line"><tt class="py-docstring">    """</tt> </tt>
<a name="L412"></a><tt class="py-lineno">412</tt>  <tt class="py-line">    <tt class="py-keyword">return</tt> <tt class="py-name">gammaln</tt><tt class="py-op">(</tt><tt class="py-name">N</tt><tt class="py-op">+</tt><tt class="py-number">1</tt><tt class="py-op">)</tt> <tt class="py-op">-</tt> <tt class="py-name">gammaln</tt><tt class="py-op">(</tt><tt class="py-name">k</tt><tt class="py-op">+</tt><tt class="py-number">1</tt><tt class="py-op">)</tt> <tt class="py-op">-</tt> <tt class="py-name">gammaln</tt><tt class="py-op">(</tt><tt class="py-name">N</tt><tt class="py-op">-</tt><tt class="py-name">k</tt><tt class="py-op">+</tt><tt class="py-number">1</tt><tt class="py-op">)</tt> </tt>
</div><a name="L413"></a><tt class="py-lineno">413</tt>  <tt class="py-line"> </tt>
<a name="_cached_projection"></a><div id="_cached_projection-def"><a name="L414"></a><tt class="py-lineno">414</tt> <a class="py-toggle" href="#" id="_cached_projection-toggle" onclick="return toggle('_cached_projection');">-</a><tt class="py-line"><tt class="py-keyword">def</tt> <a class="py-def-name" href="dadi.Numerics-module.html#_cached_projection">_cached_projection</a><tt class="py-op">(</tt><tt class="py-param">proj_to</tt><tt class="py-op">,</tt> <tt class="py-param">proj_from</tt><tt class="py-op">,</tt> <tt class="py-param">hits</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
</div><div id="_cached_projection-collapsed" style="display:none;" pad="+++" indent="++++"></div><div id="_cached_projection-expanded"><a name="L415"></a><tt class="py-lineno">415</tt>  <tt class="py-line">    <tt class="py-docstring">"""</tt> </tt>
<a name="L416"></a><tt class="py-lineno">416</tt>  <tt class="py-line"><tt class="py-docstring">    Coefficients for projection from a different fs size.</tt> </tt>
<a name="L417"></a><tt class="py-lineno">417</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L418"></a><tt class="py-lineno">418</tt>  <tt class="py-line"><tt class="py-docstring">    proj_to: Numper of samples to project down to.</tt> </tt>
<a name="L419"></a><tt class="py-lineno">419</tt>  <tt class="py-line"><tt class="py-docstring">    proj_from: Numper of samples to project from.</tt> </tt>
<a name="L420"></a><tt class="py-lineno">420</tt>  <tt class="py-line"><tt class="py-docstring">    hits: Number of derived alleles projecting from.</tt> </tt>
<a name="L421"></a><tt class="py-lineno">421</tt>  <tt class="py-line"><tt class="py-docstring">    """</tt> </tt>
<a name="L422"></a><tt class="py-lineno">422</tt>  <tt class="py-line">    <tt class="py-name">key</tt> <tt class="py-op">=</tt> <tt class="py-op">(</tt><tt class="py-name">proj_to</tt><tt class="py-op">,</tt> <tt class="py-name">proj_from</tt><tt class="py-op">,</tt> <tt class="py-name">hits</tt><tt class="py-op">)</tt> </tt>
<a name="L423"></a><tt class="py-lineno">423</tt>  <tt class="py-line">    <tt class="py-keyword">try</tt><tt class="py-op">:</tt> </tt>
<a name="L424"></a><tt class="py-lineno">424</tt>  <tt class="py-line">        <tt class="py-keyword">return</tt> <tt id="link-15" class="py-name"><a title="dadi.Numerics._projection_cache" class="py-name" href="#" onclick="return doclink('link-15', '_projection_cache', 'link-14');">_projection_cache</a></tt><tt class="py-op">[</tt><tt class="py-name">key</tt><tt class="py-op">]</tt> </tt>
<a name="L425"></a><tt class="py-lineno">425</tt>  <tt class="py-line">    <tt class="py-keyword">except</tt> <tt class="py-name">KeyError</tt><tt class="py-op">:</tt> </tt>
<a name="L426"></a><tt class="py-lineno">426</tt>  <tt class="py-line">        <tt class="py-keyword">pass</tt> </tt>
<a name="L427"></a><tt class="py-lineno">427</tt>  <tt class="py-line"> </tt>
<a name="L428"></a><tt class="py-lineno">428</tt>  <tt class="py-line">    <tt class="py-keyword">if</tt> <tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">isscalar</tt><tt class="py-op">(</tt><tt class="py-name">proj_to</tt><tt class="py-op">)</tt> <tt class="py-keyword">and</tt> <tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">isscalar</tt><tt class="py-op">(</tt><tt class="py-name">proj_from</tt><tt class="py-op">)</tt>\ </tt>
<a name="L429"></a><tt class="py-lineno">429</tt>  <tt class="py-line">       <tt class="py-keyword">and</tt> <tt class="py-name">proj_from</tt> <tt class="py-op">&lt;</tt> <tt class="py-name">proj_to</tt><tt class="py-op">:</tt> </tt>
<a name="L430"></a><tt class="py-lineno">430</tt>  <tt class="py-line">        <tt class="py-comment"># Short-circuit calculation.</tt> </tt>
<a name="L431"></a><tt class="py-lineno">431</tt>  <tt class="py-line">        <tt class="py-name">contrib</tt> <tt class="py-op">=</tt> <tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">zeros</tt><tt class="py-op">(</tt><tt class="py-name">proj_to</tt><tt class="py-op">+</tt><tt class="py-number">1</tt><tt class="py-op">)</tt> </tt>
<a name="L432"></a><tt class="py-lineno">432</tt>  <tt class="py-line">    <tt class="py-keyword">else</tt><tt class="py-op">:</tt> </tt>
<a name="L433"></a><tt class="py-lineno">433</tt>  <tt class="py-line">        <tt class="py-comment"># We set numpy's error reporting so that it will ignore underflows, </tt> </tt>
<a name="L434"></a><tt class="py-lineno">434</tt>  <tt class="py-line">        <tt class="py-comment"># because those just imply that contrib is 0.</tt> </tt>
<a name="L435"></a><tt class="py-lineno">435</tt>  <tt class="py-line">        <tt class="py-name">previous_err_state</tt> <tt class="py-op">=</tt> <tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">seterr</tt><tt class="py-op">(</tt><tt class="py-name">under</tt><tt class="py-op">=</tt><tt class="py-string">'ignore'</tt><tt class="py-op">,</tt> <tt class="py-name">divide</tt><tt class="py-op">=</tt><tt class="py-string">'raise'</tt><tt class="py-op">,</tt> </tt>
<a name="L436"></a><tt class="py-lineno">436</tt>  <tt class="py-line">                                          <tt class="py-name">over</tt><tt class="py-op">=</tt><tt class="py-string">'raise'</tt><tt class="py-op">,</tt> <tt class="py-name">invalid</tt><tt class="py-op">=</tt><tt class="py-string">'raise'</tt><tt class="py-op">)</tt> </tt>
<a name="L437"></a><tt class="py-lineno">437</tt>  <tt class="py-line">        <tt class="py-name">proj_hits</tt> <tt class="py-op">=</tt> <tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">arange</tt><tt class="py-op">(</tt><tt class="py-name">proj_to</tt><tt class="py-op">+</tt><tt class="py-number">1</tt><tt class="py-op">)</tt> </tt>
<a name="L438"></a><tt class="py-lineno">438</tt>  <tt class="py-line">        <tt class="py-comment"># For large sample sizes, we need to do the calculation in logs, and it</tt> </tt>
<a name="L439"></a><tt class="py-lineno">439</tt>  <tt class="py-line">        <tt class="py-comment"># is accurate enough for small sizes as well.</tt> </tt>
<a name="L440"></a><tt class="py-lineno">440</tt>  <tt class="py-line">        <tt class="py-name">lncontrib</tt> <tt class="py-op">=</tt> <tt id="link-16" class="py-name" targets="Function dadi.Numerics._lncomb()=dadi.Numerics-module.html#_lncomb"><a title="dadi.Numerics._lncomb" class="py-name" href="#" onclick="return doclink('link-16', '_lncomb', 'link-16');">_lncomb</a></tt><tt class="py-op">(</tt><tt class="py-name">proj_to</tt><tt class="py-op">,</tt><tt class="py-name">proj_hits</tt><tt class="py-op">)</tt> </tt>
<a name="L441"></a><tt class="py-lineno">441</tt>  <tt class="py-line">        <tt class="py-name">lncontrib</tt> <tt class="py-op">+=</tt> <tt id="link-17" class="py-name"><a title="dadi.Numerics._lncomb" class="py-name" href="#" onclick="return doclink('link-17', '_lncomb', 'link-16');">_lncomb</a></tt><tt class="py-op">(</tt><tt class="py-name">proj_from</tt><tt class="py-op">-</tt><tt class="py-name">proj_to</tt><tt class="py-op">,</tt><tt class="py-name">hits</tt><tt class="py-op">-</tt><tt class="py-name">proj_hits</tt><tt class="py-op">)</tt> </tt>
<a name="L442"></a><tt class="py-lineno">442</tt>  <tt class="py-line">        <tt class="py-name">lncontrib</tt> <tt class="py-op">-=</tt> <tt id="link-18" class="py-name"><a title="dadi.Numerics._lncomb" class="py-name" href="#" onclick="return doclink('link-18', '_lncomb', 'link-16');">_lncomb</a></tt><tt class="py-op">(</tt><tt class="py-name">proj_from</tt><tt class="py-op">,</tt> <tt class="py-name">hits</tt><tt class="py-op">)</tt> </tt>
<a name="L443"></a><tt class="py-lineno">443</tt>  <tt class="py-line">        <tt class="py-name">contrib</tt> <tt class="py-op">=</tt> <tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">exp</tt><tt class="py-op">(</tt><tt class="py-name">lncontrib</tt><tt class="py-op">)</tt> </tt>
<a name="L444"></a><tt class="py-lineno">444</tt>  <tt class="py-line">        <tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">seterr</tt><tt class="py-op">(</tt><tt class="py-op">**</tt><tt class="py-name">previous_err_state</tt><tt class="py-op">)</tt> </tt>
<a name="L445"></a><tt class="py-lineno">445</tt>  <tt class="py-line">    <tt id="link-19" class="py-name"><a title="dadi.Numerics._projection_cache" class="py-name" href="#" onclick="return doclink('link-19', '_projection_cache', 'link-14');">_projection_cache</a></tt><tt class="py-op">[</tt><tt class="py-name">key</tt><tt class="py-op">]</tt> <tt class="py-op">=</tt> <tt class="py-name">contrib</tt> </tt>
<a name="L446"></a><tt class="py-lineno">446</tt>  <tt class="py-line">    <tt class="py-keyword">return</tt> <tt class="py-name">contrib</tt> </tt>
</div><a name="L447"></a><tt class="py-lineno">447</tt>  <tt class="py-line"> </tt>
<a name="array_from_file"></a><div id="array_from_file-def"><a name="L448"></a><tt class="py-lineno">448</tt> <a class="py-toggle" href="#" id="array_from_file-toggle" onclick="return toggle('array_from_file');">-</a><tt class="py-line"><tt class="py-keyword">def</tt> <a class="py-def-name" href="dadi.Numerics-module.html#array_from_file">array_from_file</a><tt class="py-op">(</tt><tt class="py-param">fid</tt><tt class="py-op">,</tt> <tt class="py-param">return_comments</tt><tt class="py-op">=</tt><tt class="py-name">False</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
</div><div id="array_from_file-collapsed" style="display:none;" pad="+++" indent="++++"></div><div id="array_from_file-expanded"><a name="L449"></a><tt class="py-lineno">449</tt>  <tt class="py-line">    <tt class="py-docstring">"""</tt> </tt>
<a name="L450"></a><tt class="py-lineno">450</tt>  <tt class="py-line"><tt class="py-docstring">    Read array from file.</tt> </tt>
<a name="L451"></a><tt class="py-lineno">451</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L452"></a><tt class="py-lineno">452</tt>  <tt class="py-line"><tt class="py-docstring">    fid: string with file name to read from or an open file object.</tt> </tt>
<a name="L453"></a><tt class="py-lineno">453</tt>  <tt class="py-line"><tt class="py-docstring">    return_comments: If True, the return value is (fs, comments), where</tt> </tt>
<a name="L454"></a><tt class="py-lineno">454</tt>  <tt class="py-line"><tt class="py-docstring">                     comments is a list of strings containing the comments</tt> </tt>
<a name="L455"></a><tt class="py-lineno">455</tt>  <tt class="py-line"><tt class="py-docstring">                     from the file (without #'s).</tt> </tt>
<a name="L456"></a><tt class="py-lineno">456</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L457"></a><tt class="py-lineno">457</tt>  <tt class="py-line"><tt class="py-docstring">    The file format is:</tt> </tt>
<a name="L458"></a><tt class="py-lineno">458</tt>  <tt class="py-line"><tt class="py-docstring">        # Any number of comment lines beginning with a '#'</tt> </tt>
<a name="L459"></a><tt class="py-lineno">459</tt>  <tt class="py-line"><tt class="py-docstring">        A single line containing N integers giving the dimensions of the fs</tt> </tt>
<a name="L460"></a><tt class="py-lineno">460</tt>  <tt class="py-line"><tt class="py-docstring">          array. So this line would be '5 5 3' for an SFS that was 5x5x3.</tt> </tt>
<a name="L461"></a><tt class="py-lineno">461</tt>  <tt class="py-line"><tt class="py-docstring">          (That would be 4x4x2 *samples*.)</tt> </tt>
<a name="L462"></a><tt class="py-lineno">462</tt>  <tt class="py-line"><tt class="py-docstring">        A single line giving the array elements. The order of elements is </tt> </tt>
<a name="L463"></a><tt class="py-lineno">463</tt>  <tt class="py-line"><tt class="py-docstring">          e.g.: fs[0,0,0] fs[0,0,1] fs[0,0,2] ... fs[0,1,0] fs[0,1,1] ...</tt> </tt>
<a name="L464"></a><tt class="py-lineno">464</tt>  <tt class="py-line"><tt class="py-docstring">    """</tt> </tt>
<a name="L465"></a><tt class="py-lineno">465</tt>  <tt class="py-line">    <tt class="py-name">newfile</tt> <tt class="py-op">=</tt> <tt class="py-name">False</tt> </tt>
<a name="L466"></a><tt class="py-lineno">466</tt>  <tt class="py-line">    <tt class="py-comment"># Try to read from fid. If we can't, assume it's something that we can</tt> </tt>
<a name="L467"></a><tt class="py-lineno">467</tt>  <tt class="py-line">    <tt class="py-comment"># use to open a file.</tt> </tt>
<a name="L468"></a><tt class="py-lineno">468</tt>  <tt class="py-line">    <tt class="py-keyword">if</tt> <tt class="py-keyword">not</tt> <tt class="py-name">hasattr</tt><tt class="py-op">(</tt><tt class="py-name">fid</tt><tt class="py-op">,</tt> <tt class="py-string">'read'</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
<a name="L469"></a><tt class="py-lineno">469</tt>  <tt class="py-line">        <tt class="py-name">newfile</tt> <tt class="py-op">=</tt> <tt class="py-name">True</tt> </tt>
<a name="L470"></a><tt class="py-lineno">470</tt>  <tt class="py-line">        <tt class="py-name">fid</tt> <tt class="py-op">=</tt> <tt class="py-name">file</tt><tt class="py-op">(</tt><tt class="py-name">fid</tt><tt class="py-op">,</tt> <tt class="py-string">'r'</tt><tt class="py-op">)</tt> </tt>
<a name="L471"></a><tt class="py-lineno">471</tt>  <tt class="py-line"> </tt>
<a name="L472"></a><tt class="py-lineno">472</tt>  <tt class="py-line">    <tt class="py-name">line</tt> <tt class="py-op">=</tt> <tt class="py-name">fid</tt><tt class="py-op">.</tt><tt class="py-name">readline</tt><tt class="py-op">(</tt><tt class="py-op">)</tt> </tt>
<a name="L473"></a><tt class="py-lineno">473</tt>  <tt class="py-line">    <tt class="py-comment"># Strip out the comments</tt> </tt>
<a name="L474"></a><tt class="py-lineno">474</tt>  <tt class="py-line">    <tt class="py-name">comments</tt> <tt class="py-op">=</tt> <tt class="py-op">[</tt><tt class="py-op">]</tt> </tt>
<a name="L475"></a><tt class="py-lineno">475</tt>  <tt class="py-line">    <tt class="py-keyword">while</tt> <tt class="py-name">line</tt><tt class="py-op">.</tt><tt class="py-name">startswith</tt><tt class="py-op">(</tt><tt class="py-string">'#'</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
<a name="L476"></a><tt class="py-lineno">476</tt>  <tt class="py-line">        <tt class="py-name">comments</tt><tt class="py-op">.</tt><tt class="py-name">append</tt><tt class="py-op">(</tt><tt class="py-name">line</tt><tt class="py-op">[</tt><tt class="py-number">1</tt><tt class="py-op">:</tt><tt class="py-op">]</tt><tt class="py-op">.</tt><tt class="py-name">strip</tt><tt class="py-op">(</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> </tt>
<a name="L477"></a><tt class="py-lineno">477</tt>  <tt class="py-line">        <tt class="py-name">line</tt> <tt class="py-op">=</tt> <tt class="py-name">fid</tt><tt class="py-op">.</tt><tt class="py-name">readline</tt><tt class="py-op">(</tt><tt class="py-op">)</tt> </tt>
<a name="L478"></a><tt class="py-lineno">478</tt>  <tt class="py-line"> </tt>
<a name="L479"></a><tt class="py-lineno">479</tt>  <tt class="py-line">    <tt class="py-comment"># Read the shape of the data</tt> </tt>
<a name="L480"></a><tt class="py-lineno">480</tt>  <tt class="py-line">    <tt class="py-name">shape</tt> <tt class="py-op">=</tt> <tt class="py-name">tuple</tt><tt class="py-op">(</tt><tt class="py-op">[</tt><tt class="py-name">int</tt><tt class="py-op">(</tt><tt class="py-name">d</tt><tt class="py-op">)</tt> <tt class="py-keyword">for</tt> <tt class="py-name">d</tt> <tt class="py-keyword">in</tt> <tt class="py-name">line</tt><tt class="py-op">.</tt><tt class="py-name">split</tt><tt class="py-op">(</tt><tt class="py-op">)</tt><tt class="py-op">]</tt><tt class="py-op">)</tt> </tt>
<a name="L481"></a><tt class="py-lineno">481</tt>  <tt class="py-line"> </tt>
<a name="L482"></a><tt class="py-lineno">482</tt>  <tt class="py-line">    <tt class="py-name">data</tt> <tt class="py-op">=</tt> <tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">fromfile</tt><tt class="py-op">(</tt><tt class="py-name">fid</tt><tt class="py-op">,</tt> <tt class="py-name">count</tt><tt class="py-op">=</tt><tt class="py-name">numpy</tt><tt class="py-op">.</tt><tt class="py-name">product</tt><tt class="py-op">(</tt><tt class="py-name">shape</tt><tt class="py-op">)</tt><tt class="py-op">,</tt> <tt class="py-name">sep</tt><tt class="py-op">=</tt><tt class="py-string">' '</tt><tt class="py-op">)</tt> </tt>
<a name="L483"></a><tt class="py-lineno">483</tt>  <tt class="py-line">    <tt class="py-comment"># fromfile returns a 1-d array. Reshape it to the proper form.</tt> </tt>
<a name="L484"></a><tt class="py-lineno">484</tt>  <tt class="py-line">    <tt class="py-name">data</tt> <tt class="py-op">=</tt> <tt class="py-name">data</tt><tt class="py-op">.</tt><tt class="py-name">reshape</tt><tt class="py-op">(</tt><tt class="py-op">*</tt><tt class="py-name">shape</tt><tt class="py-op">)</tt> </tt>
<a name="L485"></a><tt class="py-lineno">485</tt>  <tt class="py-line"> </tt>
<a name="L486"></a><tt class="py-lineno">486</tt>  <tt class="py-line">    <tt class="py-comment"># If we opened a new file, clean it up.</tt> </tt>
<a name="L487"></a><tt class="py-lineno">487</tt>  <tt class="py-line">    <tt class="py-keyword">if</tt> <tt class="py-name">newfile</tt><tt class="py-op">:</tt> </tt>
<a name="L488"></a><tt class="py-lineno">488</tt>  <tt class="py-line">        <tt class="py-name">fid</tt><tt class="py-op">.</tt><tt class="py-name">close</tt><tt class="py-op">(</tt><tt class="py-op">)</tt> </tt>
<a name="L489"></a><tt class="py-lineno">489</tt>  <tt class="py-line"> </tt>
<a name="L490"></a><tt class="py-lineno">490</tt>  <tt class="py-line">    <tt class="py-keyword">if</tt> <tt class="py-keyword">not</tt> <tt class="py-name">return_comments</tt><tt class="py-op">:</tt> </tt>
<a name="L491"></a><tt class="py-lineno">491</tt>  <tt class="py-line">        <tt class="py-keyword">return</tt> <tt class="py-name">data</tt> </tt>
<a name="L492"></a><tt class="py-lineno">492</tt>  <tt class="py-line">    <tt class="py-keyword">else</tt><tt class="py-op">:</tt> </tt>
<a name="L493"></a><tt class="py-lineno">493</tt>  <tt class="py-line">        <tt class="py-keyword">return</tt> <tt class="py-name">data</tt><tt class="py-op">,</tt><tt class="py-name">comments</tt> </tt>
</div><a name="L494"></a><tt class="py-lineno">494</tt>  <tt class="py-line"> </tt>
<a name="array_to_file"></a><div id="array_to_file-def"><a name="L495"></a><tt class="py-lineno">495</tt> <a class="py-toggle" href="#" id="array_to_file-toggle" onclick="return toggle('array_to_file');">-</a><tt class="py-line"><tt class="py-keyword">def</tt> <a class="py-def-name" href="dadi.Numerics-module.html#array_to_file">array_to_file</a><tt class="py-op">(</tt><tt class="py-param">data</tt><tt class="py-op">,</tt> <tt class="py-param">fid</tt><tt class="py-op">,</tt> <tt class="py-param">precision</tt><tt class="py-op">=</tt><tt class="py-number">16</tt><tt class="py-op">,</tt> <tt class="py-param">comment_lines</tt> <tt class="py-op">=</tt> <tt class="py-op">[</tt><tt class="py-op">]</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
</div><div id="array_to_file-collapsed" style="display:none;" pad="+++" indent="++++"></div><div id="array_to_file-expanded"><a name="L496"></a><tt class="py-lineno">496</tt>  <tt class="py-line">    <tt class="py-docstring">"""</tt> </tt>
<a name="L497"></a><tt class="py-lineno">497</tt>  <tt class="py-line"><tt class="py-docstring">    Write array to file.</tt> </tt>
<a name="L498"></a><tt class="py-lineno">498</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L499"></a><tt class="py-lineno">499</tt>  <tt class="py-line"><tt class="py-docstring">    data: array to write</tt> </tt>
<a name="L500"></a><tt class="py-lineno">500</tt>  <tt class="py-line"><tt class="py-docstring">    fid: string with file name to write to or an open file object.</tt> </tt>
<a name="L501"></a><tt class="py-lineno">501</tt>  <tt class="py-line"><tt class="py-docstring">    precision: precision with which to write out entries of the SFS. (They </tt> </tt>
<a name="L502"></a><tt class="py-lineno">502</tt>  <tt class="py-line"><tt class="py-docstring">               are formated via %.&lt;p&gt;g, where &lt;p&gt; is the precision.)</tt> </tt>
<a name="L503"></a><tt class="py-lineno">503</tt>  <tt class="py-line"><tt class="py-docstring">    comment lines: list of strings to be used as comment lines in the header</tt> </tt>
<a name="L504"></a><tt class="py-lineno">504</tt>  <tt class="py-line"><tt class="py-docstring">                   of the output file.</tt> </tt>
<a name="L505"></a><tt class="py-lineno">505</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L506"></a><tt class="py-lineno">506</tt>  <tt class="py-line"><tt class="py-docstring">    The file format is:</tt> </tt>
<a name="L507"></a><tt class="py-lineno">507</tt>  <tt class="py-line"><tt class="py-docstring">        # Any number of comment lines beginning with a '#'</tt> </tt>
<a name="L508"></a><tt class="py-lineno">508</tt>  <tt class="py-line"><tt class="py-docstring">        A single line containing N integers giving the dimensions of the fs</tt> </tt>
<a name="L509"></a><tt class="py-lineno">509</tt>  <tt class="py-line"><tt class="py-docstring">          array. So this line would be '5 5 3' for an SFS that was 5x5x3.</tt> </tt>
<a name="L510"></a><tt class="py-lineno">510</tt>  <tt class="py-line"><tt class="py-docstring">          (That would be 4x4x2 *samples*.)</tt> </tt>
<a name="L511"></a><tt class="py-lineno">511</tt>  <tt class="py-line"><tt class="py-docstring">        A single line giving the array elements. The order of elements is </tt> </tt>
<a name="L512"></a><tt class="py-lineno">512</tt>  <tt class="py-line"><tt class="py-docstring">          e.g.: fs[0,0,0] fs[0,0,1] fs[0,0,2] ... fs[0,1,0] fs[0,1,1] ...</tt> </tt>
<a name="L513"></a><tt class="py-lineno">513</tt>  <tt class="py-line"><tt class="py-docstring">    """</tt> </tt>
<a name="L514"></a><tt class="py-lineno">514</tt>  <tt class="py-line">    <tt class="py-comment"># Open the file object.</tt> </tt>
<a name="L515"></a><tt class="py-lineno">515</tt>  <tt class="py-line">    <tt class="py-name">newfile</tt> <tt class="py-op">=</tt> <tt class="py-name">False</tt> </tt>
<a name="L516"></a><tt class="py-lineno">516</tt>  <tt class="py-line">    <tt class="py-keyword">if</tt> <tt class="py-keyword">not</tt> <tt class="py-name">hasattr</tt><tt class="py-op">(</tt><tt class="py-name">fid</tt><tt class="py-op">,</tt> <tt class="py-string">'write'</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
<a name="L517"></a><tt class="py-lineno">517</tt>  <tt class="py-line">        <tt class="py-name">newfile</tt> <tt class="py-op">=</tt> <tt class="py-name">True</tt> </tt>
<a name="L518"></a><tt class="py-lineno">518</tt>  <tt class="py-line">        <tt class="py-name">fid</tt> <tt class="py-op">=</tt> <tt class="py-name">file</tt><tt class="py-op">(</tt><tt class="py-name">fid</tt><tt class="py-op">,</tt> <tt class="py-string">'w'</tt><tt class="py-op">)</tt> </tt>
<a name="L519"></a><tt class="py-lineno">519</tt>  <tt class="py-line"> </tt>
<a name="L520"></a><tt class="py-lineno">520</tt>  <tt class="py-line">    <tt class="py-comment"># Write comments</tt> </tt>
<a name="L521"></a><tt class="py-lineno">521</tt>  <tt class="py-line">    <tt class="py-keyword">for</tt> <tt class="py-name">line</tt> <tt class="py-keyword">in</tt> <tt class="py-name">comment_lines</tt><tt class="py-op">:</tt> </tt>
<a name="L522"></a><tt class="py-lineno">522</tt>  <tt class="py-line">        <tt class="py-name">fid</tt><tt class="py-op">.</tt><tt class="py-name">write</tt><tt class="py-op">(</tt><tt class="py-string">'# '</tt><tt class="py-op">)</tt> </tt>
<a name="L523"></a><tt class="py-lineno">523</tt>  <tt class="py-line">        <tt class="py-name">fid</tt><tt class="py-op">.</tt><tt class="py-name">write</tt><tt class="py-op">(</tt><tt class="py-name">line</tt><tt class="py-op">.</tt><tt class="py-name">strip</tt><tt class="py-op">(</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> </tt>
<a name="L524"></a><tt class="py-lineno">524</tt>  <tt class="py-line">        <tt class="py-name">fid</tt><tt class="py-op">.</tt><tt class="py-name">write</tt><tt class="py-op">(</tt><tt class="py-name">os</tt><tt class="py-op">.</tt><tt class="py-name">linesep</tt><tt class="py-op">)</tt> </tt>
<a name="L525"></a><tt class="py-lineno">525</tt>  <tt class="py-line"> </tt>
<a name="L526"></a><tt class="py-lineno">526</tt>  <tt class="py-line">    <tt class="py-comment"># Write out the shape of the fs</tt> </tt>
<a name="L527"></a><tt class="py-lineno">527</tt>  <tt class="py-line">    <tt class="py-keyword">for</tt> <tt class="py-name">elem</tt> <tt class="py-keyword">in</tt> <tt class="py-name">data</tt><tt class="py-op">.</tt><tt class="py-name">shape</tt><tt class="py-op">:</tt> </tt>
<a name="L528"></a><tt class="py-lineno">528</tt>  <tt class="py-line">        <tt class="py-name">fid</tt><tt class="py-op">.</tt><tt class="py-name">write</tt><tt class="py-op">(</tt><tt class="py-string">'%i '</tt> <tt class="py-op">%</tt> <tt class="py-name">elem</tt><tt class="py-op">)</tt> </tt>
<a name="L529"></a><tt class="py-lineno">529</tt>  <tt class="py-line">    <tt class="py-name">fid</tt><tt class="py-op">.</tt><tt class="py-name">write</tt><tt class="py-op">(</tt><tt class="py-name">os</tt><tt class="py-op">.</tt><tt class="py-name">linesep</tt><tt class="py-op">)</tt> </tt>
<a name="L530"></a><tt class="py-lineno">530</tt>  <tt class="py-line"> </tt>
<a name="L531"></a><tt class="py-lineno">531</tt>  <tt class="py-line">    <tt class="py-keyword">if</tt> <tt class="py-name">hasattr</tt><tt class="py-op">(</tt><tt class="py-name">data</tt><tt class="py-op">,</tt> <tt class="py-string">'filled'</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
<a name="L532"></a><tt class="py-lineno">532</tt>  <tt class="py-line">        <tt class="py-comment"># Masked entries in the fs will go in as 'nan'</tt> </tt>
<a name="L533"></a><tt class="py-lineno">533</tt>  <tt class="py-line">        <tt class="py-name">data</tt> <tt class="py-op">=</tt> <tt class="py-name">data</tt><tt class="py-op">.</tt><tt class="py-name">filled</tt><tt class="py-op">(</tt><tt class="py-op">)</tt> </tt>
<a name="L534"></a><tt class="py-lineno">534</tt>  <tt class="py-line">    <tt class="py-comment"># Write to file</tt> </tt>
<a name="L535"></a><tt class="py-lineno">535</tt>  <tt class="py-line">    <tt class="py-name">data</tt><tt class="py-op">.</tt><tt class="py-name">tofile</tt><tt class="py-op">(</tt><tt class="py-name">fid</tt><tt class="py-op">,</tt> <tt class="py-string">' '</tt><tt class="py-op">,</tt> <tt class="py-string">'%%.%ig'</tt> <tt class="py-op">%</tt> <tt class="py-name">precision</tt><tt class="py-op">)</tt> </tt>
<a name="L536"></a><tt class="py-lineno">536</tt>  <tt class="py-line">    <tt class="py-name">fid</tt><tt class="py-op">.</tt><tt class="py-name">write</tt><tt class="py-op">(</tt><tt class="py-name">os</tt><tt class="py-op">.</tt><tt class="py-name">linesep</tt><tt class="py-op">)</tt> </tt>
<a name="L537"></a><tt class="py-lineno">537</tt>  <tt class="py-line"> </tt>
<a name="L538"></a><tt class="py-lineno">538</tt>  <tt class="py-line">    <tt class="py-comment"># Close file</tt> </tt>
<a name="L539"></a><tt class="py-lineno">539</tt>  <tt class="py-line">    <tt class="py-keyword">if</tt> <tt class="py-name">newfile</tt><tt class="py-op">:</tt> </tt>
<a name="L540"></a><tt class="py-lineno">540</tt>  <tt class="py-line">        <tt class="py-name">fid</tt><tt class="py-op">.</tt><tt class="py-name">close</tt><tt class="py-op">(</tt><tt class="py-op">)</tt> </tt>
</div><a name="L541"></a><tt class="py-lineno">541</tt>  <tt class="py-line"> </tt><script type="text/javascript">
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